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Zubov’s Method for Differential Games
Lars Grune
Mathematisches InstitutUniversitat Bayreuth
Joint work with Oana Silvia Serea,Ecole Polytechnique, Palaiseau, France
International Workshop “The Dynamics of Control”Irsee, 1st–3rd October, 2010
Happy Birthday Fritz!
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Zubov’s Method for Differential Games
Lars Grune
Mathematisches InstitutUniversitat Bayreuth
Joint work with Oana Silvia Serea,Ecole Polytechnique, Palaiseau, France
International Workshop “The Dynamics of Control”Irsee, 1st–3rd October, 2010
Happy Birthday Fritz!
![Page 3: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/3.jpg)
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd
(without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
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Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
![Page 5: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/5.jpg)
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
![Page 6: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/6.jpg)
Introduction: The Uncontrolled CaseConsider the autonomous ODE
x(t) = f(x(t)), x ∈ Rd
with solutions Φ(t, x0) and locally exponentially stableequilibrium x∗ ∈ Rd (without loss of generality x∗ = 0)
i.e., there exists a neighborhood N of x∗ = 0 and constantsc, σ > 0, such that for all x0 ∈ N :
‖Φ(t, x0)‖ ≤ ce−σt‖x0‖
Problem: What is the domain of attraction
D := {x ∈ Rd |Φ(t, x)→ x∗ = 0} ?
Lars Grune, Zubov’s Method for Differential Games, p. 2 of 23
![Page 7: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/7.jpg)
Example for a Domain of AttractionFluid Dynamics: Explanation of the difference between linearstability and experimental instability for large Reynoldsnumbers [Trefethen et al., Science, 1993]
Lars Grune, Zubov’s Method for Differential Games, p. 3 of 23
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Zubov’s Equation [1964]For a continuous function h : Rd → R≥0 withh(x) = 0⇔ x = x∗ consider the PDE “Zubov’s Equation”
Dw(x) · f(x) = −h(x)(1− w(x))
with w : Rd → R and boundary condition w(x∗) = 0
Then: under suitable conditions on h this equation has aunique solution w : Rd → [0, 1] with
w(x) = 0 ⇔ x = x∗
and D satisfies the level set characterization
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Lars Grune, Zubov’s Method for Differential Games, p. 4 of 23
![Page 9: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/9.jpg)
Zubov’s Equation [1964]For a continuous function h : Rd → R≥0 withh(x) = 0⇔ x = x∗ consider the PDE “Zubov’s Equation”
Dw(x) · f(x) = −h(x)(1− w(x))
with w : Rd → R and boundary condition w(x∗) = 0
Then: under suitable conditions on h this equation has aunique solution w : Rd → [0, 1] with
w(x) = 0 ⇔ x = x∗
and D satisfies the level set characterization
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Lars Grune, Zubov’s Method for Differential Games, p. 4 of 23
![Page 10: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/10.jpg)
Example
x1(t) = −x1(t) + x1(t)3, x2(t) = −x2(t) + x2(t)3
D = [−1, 1]2
, h(x) = 5‖x‖2
−10
1
−1
0
1
0
0.2
0.4
0.6
0.8
1
x1
x2
v
Lars Grune, Zubov’s Method for Differential Games, p. 5 of 23
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Example
x1(t) = −x1(t) + x1(t)3, x2(t) = −x2(t) + x2(t)3
D = [−1, 1]2, h(x) = 5‖x‖2
−10
1
−1
0
1
0
0.2
0.4
0.6
0.8
1
x1
x2
v
Lars Grune, Zubov’s Method for Differential Games, p. 5 of 23
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Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
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Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
![Page 14: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/14.jpg)
Integral Equation
Dw(x) · f(x) = −h(x)(1− w(x))
D = w−1([0, 1)) := {x ∈ Rd |w(x) ∈ [0, 1)}
Why does this characterization hold?
Integration of Zubov’s equation and subsequentintegration by parts yields the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
Thus:
Φ(t, x)→ x∗ ⇔∫ ∞
0
h(Φ(t, x))dt <∞ ⇔ w(x) < 1
Lars Grune, Zubov’s Method for Differential Games, p. 6 of 23
![Page 15: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/15.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction D
an existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 16: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/16.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 17: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/17.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 18: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/18.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 19: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/19.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 20: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/20.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
![Page 21: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/21.jpg)
Zubov’s Equation — DiscussionZubov’s Equation yields
a characterization of the domain of attraction Dan existence result for a Lyapunov function v on D, i.e.,on the the largest possible domain
an — in principle — constructive method for thecomputation of v and D — analytically or numerically
additional insight through PDE formulation
Generalizations exist, e.g., for
periodic orbits [Aulbach ’83]
perturbed systems (deterministic: [Camilli, Gr., Wirth ’01],
stochastic: [Camilli, Loreti ’06; Camilli, Gr. ’03])
control systems (deterministic: [Sontag ’83, Camilli, Gr.,
Wirth ’08], stochastic: [Camilli, Cesaroni, Gr., Wirth ’06])
Lars Grune, Zubov’s Method for Differential Games, p. 7 of 23
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Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
![Page 23: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/23.jpg)
Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
![Page 24: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/24.jpg)
Control and PerturbationIn this talk we consider generalizations of this method forcontrolled and deterministically perturbed systems
x(t) = f(x(t), u(t), v(t))
with x(t) ∈ Rd
u ∈ U = {u : [0,∞)→ U, measurable}v ∈ V = {v : [0,∞)→ V, measurable}U ⊂ Rm, V ⊂ Rl compact
Problem: stabilization under uncertainty
u = control, trying to achieve Φ(t, x0, u, v)→ x∗
v = perturbation, trying to keep Φ(t, x0, u, v) away from x∗
(convergence to x∗ = 0 can be generalized to arbitrary compact sets)
Lars Grune, Zubov’s Method for Differential Games, p. 8 of 23
![Page 25: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/25.jpg)
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
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Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
![Page 27: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/27.jpg)
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
![Page 28: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/28.jpg)
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
![Page 29: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/29.jpg)
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J
zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
![Page 30: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/30.jpg)
Extension of Integral EquationRecall: Zubov’s method relies on the integral equation
w(x) = 1− e−R∞0 h(Φ(t,x))dt
For the solutions Φ(t, x, u, w) of x(t) = f(x(t), u(t), v(t)) define
J(x, u, v) = 1− e−R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
Local exponential controllability and an appropriate choice ofh : Rd × U × V → R+
0 ensure
J(x, u, v) < 1 ⇔ Φ(t, x, u, v)→ 0
J(x, u, v) = 1 ⇔ Φ(t, x, u, v) 6→ 0
Thus, u should minimize J while v should maximize J zero sum differential game (min-max problem)
Lars Grune, Zubov’s Method for Differential Games, p. 9 of 23
![Page 31: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/31.jpg)
Information Exchange between u and vWhat do u and v know about each other?
Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 32: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/32.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 33: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/33.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 34: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/34.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 35: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/35.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U
— overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 36: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/36.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V
— unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 37: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/37.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t)
— causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 38: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/38.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t]
— causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 39: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/39.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 40: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/40.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)
General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 41: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/41.jpg)
Information Exchange between u and vWhat do u and v know about each other? Possible settings:
when the perturbation chooses v : [0,∞)→ V , it knowsu : [0,∞)→ U — overly conservative, non causal
when the control chooses u : [0,∞)→ U , it knowsv : [0,∞)→ V — unrealistic, non causal
at time t, the perturbation knows u|[0,t] and the controlknows v|[0,t) — causal
at time t, the perturbation knows u|[0,t) and the controlknows v|[0,t] — causal
The last two settings are also realistic, since by Hamilton-Jacobi theory for choosing the optimal u and v knowing u|[0,t)and v|[0,t) is equivalent to knowing the solution Φ|[0,t)General question for differential games: does the“infinitesimal” advantage make a difference?
Lars Grune, Zubov’s Method for Differential Games, p. 10 of 23
![Page 42: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/42.jpg)
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
![Page 43: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/43.jpg)
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
![Page 44: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/44.jpg)
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
![Page 45: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/45.jpg)
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
![Page 46: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/46.jpg)
Formalization of the Information StructureWe formalize the last two cases by defining the set ∆ ofnonanticipative strategies for the perturbation as the set ofmaps β : U → V with the following property for all u1, u2 ∈ Uand all s > 0:
u1(τ) = u2(τ) for almost all τ ∈ [0, s]
⇒ β(u1)(τ) = β(u2)(τ) for almost all τ ∈ [0, s]
similarly, we define the set Γ of nonanticipative strategiesα : V → U for the control
upper value: w+(x) := supβ∈∆
infu∈U
J(x, u, β(u))
lower value: w−(x) := infα∈Γ
supv∈V
J(x, α(v), v)
Keep in mind: the strategy player has an infinitesimal advantage
Lars Grune, Zubov’s Method for Differential Games, p. 11 of 23
![Page 47: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/47.jpg)
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
![Page 48: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/48.jpg)
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
![Page 49: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/49.jpg)
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
![Page 50: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/50.jpg)
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖
(can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
![Page 51: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/51.jpg)
Domains of ControllabilityWe need two different domains of controllability
D+ = (w+)−1([0, 1)) and D− = (w−)−1([0, 1))
upper domain of uniform asymptotic controllability
D+ =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 such thatfor each β ∈ ∆ there existsu ∈ U with ‖Φ(t, x, u, β(u))‖ ≤ θ(t)
lower domain of uniform asymptotic controllability
D− =
x ∈ Rd
∣∣∣∣∣∣there exists θ(t)→ 0 and α ∈ Γsuch that for each v ∈ V the inequality‖Φ(t, x, α(v), v)‖ ≤ θ(t) holds
Local exponential controllability is defined analogously withθ(t) = ce−σt‖x‖ (can be generalized to uniform convergence)
Lars Grune, Zubov’s Method for Differential Games, p. 12 of 23
![Page 52: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/52.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 53: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/53.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 54: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/54.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 55: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/55.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 56: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/56.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 57: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/57.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 58: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/58.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 59: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/59.jpg)
ExampleCan the upper and lower domain be different?
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ R
x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
upper domain: perturbation chooses strategy β
set β(u)(t) := u(t)
x(t) = −x(t) + x(t)3 for all u ∈ U D+ = (−1, 1)
lower domain: control chooses strategy α
set α(v)(t) := −v(t)
x(t) = −x(t)− x(t)3 for all v ∈ V D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 13 of 23
![Page 60: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/60.jpg)
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}
Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
![Page 61: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/61.jpg)
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
![Page 62: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/62.jpg)
Formal Derivation of Zubov’s Equation
w+(x) := supβ∈∆
infu∈U
{1− e−
R∞0 h(Φ(t,x,u,v),u(t),v(t))dt
}satisfies for all T > 0 the optimality principle
w+(x) = supβ∈∆
infu∈U
{1− e−
R T0 h(Φ(t,x,u,v),u(t),v(t))dt[1− w+(Φ(T, x, u, v))]
}Division by −T and passing to the limit for T → 0 yields theHamilton–Jacobi–Isaacs equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
“generalized Zubov Equation”
Lars Grune, Zubov’s Method for Differential Games, p. 14 of 23
![Page 63: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/63.jpg)
Formal Derivation of Zubov’s Equationw+ formally satisfies the generalized Zubov equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
Likewise, w− formally satisfies the generalized Zubov equation
H−(x,w−(x), Dw−(x)) = 0
with Hamiltonian
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 15 of 23
![Page 64: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/64.jpg)
Formal Derivation of Zubov’s Equationw+ formally satisfies the generalized Zubov equation
H+(x,w+(x), Dw+(x)) = 0
with Hamiltonian
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
Likewise, w− formally satisfies the generalized Zubov equation
H−(x,w−(x), Dw−(x)) = 0
with Hamiltonian
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 15 of 23
![Page 65: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/65.jpg)
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
![Page 66: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/66.jpg)
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
![Page 67: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/67.jpg)
Nonsmooth SolutionProblem: w+ and w− from the integral equations are typicallynonsmooth
Example: w+ for x(t) = −x(t) + u(t)v(t)x(t)3, withU = V = {−1, 1}, h(x, u, v) = x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
viscositysolution
Lars Grune, Zubov’s Method for Differential Games, p. 16 of 23
![Page 68: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/68.jpg)
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
![Page 69: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/69.jpg)
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
![Page 70: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/70.jpg)
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
![Page 71: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/71.jpg)
Viscosity Solution
Super– and subdifferential:
v
x
v
x
D+v(x) D−v(x)
w viscosity supersolution: H(x,w(x), p) ≥ 0 for all p ∈ D−w(x)
w viscosity subsolution: H(x,w(x), p) ≤ 0 for all p ∈ D+w(x)
w viscosity solution, if both holds [Crandall, Lions 82]
Lars Grune, Zubov’s Method for Differential Games, p. 17 of 23
![Page 72: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/72.jpg)
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
![Page 73: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/73.jpg)
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
![Page 74: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/74.jpg)
Existence and UniquenessWith this solution concept and with the help of “sub– andsuperoptimality principles” for viscosity super– andsubsolutions [Soravia 95] we arrive at the following Theorem:
w+ is the unique continuous viscosity solution of the equation
supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w+(0) = 0
w− is the unique continuous viscosity solution of the equation
infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)} = 0
with w−(0) = 0
Furthermore, the characterizations D+ = (w+)−1([0, 1)) andD− = (w−)−1([0, 1)) hold
Lars Grune, Zubov’s Method for Differential Games, p. 18 of 23
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ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
![Page 76: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/76.jpg)
ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
![Page 77: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/77.jpg)
ExampleConsider the example from before
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}
For h(x, u, v) = x2 we can compute explicitly
w+(x) =
{1−√
1− x2, |x| < 11, |x| ≥ 1
w−(x) =
√1 + x2 − 1√
1 + x2
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
x
W+(x
)
−5 −4 −3 −2 −1 0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
W−(x
)
This confirms D+ = (−1, 1) and D− = (−∞,∞)
Lars Grune, Zubov’s Method for Differential Games, p. 19 of 23
![Page 78: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/78.jpg)
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
![Page 79: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/79.jpg)
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
![Page 80: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/80.jpg)
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds.
Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
![Page 81: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/81.jpg)
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
![Page 82: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/82.jpg)
When does D+ = D− hold?
When does playing strategies yield no advantage?
The level set characterization implies that D+ = D− holds ifthere exists h such that w+ = w− holds.
The existence and uniqueness theorem implies that w+ = w−
holds if H+ = H− holds. Thus: H+ = H− ⇒ D+ = D−
Recall:
H+(x,w, p) = supu∈U
infv∈V{−p · f(x, u, v)− h(x, u, v)(1− w)}
H−(x,w, p) = infv∈V
supu∈U{−p · f(x, u, v)− h(x, u, v)(1− w)}
Lars Grune, Zubov’s Method for Differential Games, p. 20 of 23
![Page 83: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/83.jpg)
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
![Page 84: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/84.jpg)
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
![Page 85: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/85.jpg)
When does D+ = D− hold?
For the special case of h(x, u, v) = h(x) we get
H+(x,w, p) = H−(x,w, p)
⇔ supu∈U
infv∈V{−p · f(x, u, v)} = inf
v∈Vsupu∈U{−p · f(x, u, v)}
This condition (for all p ∈ Rn) is known as Isaacs’ condition
Theorem: Isaacs’ condition implies D+ = D−
This theorem extends a well known result from capture basinsin finite time pursuit evasion games to domains ofcontrollability of asymptotically controllable sets
Lars Grune, Zubov’s Method for Differential Games, p. 21 of 23
![Page 86: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/86.jpg)
ExampleIn our example
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}we have D+ = (−1, 1) 6= (−∞,∞) = D−
Isaacs’ condition must be violated
Indeed, for p = 1 and x = 1 we have
p · f(x, u, v) = −1 + uv
and thus
supu∈U
infv∈V{−p · f(x, u, v)} = sup
u∈Uinfv∈V{1− uv} = 0
but
infv∈V
supu∈U{−p · f(x, u, v)} = inf
v∈Vsupu∈U{1− uv} = 2
Lars Grune, Zubov’s Method for Differential Games, p. 22 of 23
![Page 87: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/87.jpg)
ExampleIn our example
x(t) = −x(t) + u(t)v(t)x(t)3, x(t) ∈ Rwith x(t) ∈ R, u(t) ∈ U = {−1, 1}, v(t) ∈ V = {−1, 1}we have D+ = (−1, 1) 6= (−∞,∞) = D−
Isaacs’ condition must be violated
Indeed, for p = 1 and x = 1 we have
p · f(x, u, v) = −1 + uv
and thus
supu∈U
infv∈V{−p · f(x, u, v)} = sup
u∈Uinfv∈V{1− uv} = 0
but
infv∈V
supu∈U{−p · f(x, u, v)} = inf
v∈Vsupu∈U{1− uv} = 2
Lars Grune, Zubov’s Method for Differential Games, p. 22 of 23
![Page 88: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/88.jpg)
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
![Page 89: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/89.jpg)
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
![Page 90: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/90.jpg)
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23
![Page 91: Zubov’s Method for Di erential Games - uni-bayreuth.denumerik.mathematik.uni-bayreuth.de/irsee_2010/download/... · 2010. 10. 8. · Zubov’s Method for Di erential Games Lars](https://reader036.vdocuments.net/reader036/viewer/2022081621/611dcefd075f561d6d16615b/html5/thumbnails/91.jpg)
Conclusions
Zubov’s method gives a characterization of the domain ofasymptotic controllability via the level set of a solution ofa first order PDE
Using viscosity solutions, the method can be extended toa differential game setting
Upper and lower value w+ and w− and the respectivedomains of controllability D+ and D− are defined andanalyzed seperately
Under the well known Isaacs condition the upper andlower domains of controllability D+ and D− coincide
Lars Grune, Zubov’s Method for Differential Games, p. 23 of 23