Advances in Dynamic Games pp 23-37

Part of the Annals of the International Society of Dynamic Games book series (AISDG, volume 8)

# Level Sweeping of the Value Function in Linear Differential Games

• Sergey S. Kumkov
• Valerii S. Patsko

## Abstract

In this chapter one considers a linear antagonistic differential game with fixed terminal time T, geometric constraints on the players’ controls, and continuous quasi-convex payoff function ϕ depending on two components xi, xj of the phase vector x. Let $$\mathcal{M}_c = \{ x:\varphi (x_1 ,x_j ) \leqslant c\}$$ be a level set (a Lebesgue set) of the payoff function. One says that the function ϕ possesses the level sweeping property if for any pair of constants c1 < c2 the relation $$\mathcal{M}_{c2} = \mathcal{M}_{c1} + (\mathcal{M}_{c2} - *\mathcal{M}_{c1} )$$ holds. Here, the symbols + and $$-$$ mean algebraic sum (Minkowski sum) and geometric difference (Minkowski difference). Let $$\mathcal{W}_c$$ be a level set of the value function $$(t,x) \mapsto \mathcal{V}(t,x)$$. The main result of this work is the proof of the fact that if the payoff function ϕ possesses the level sweeping property, then for any t ∈ [t0, T] the function $$x \mapsto \mathcal{V}(t,x)$$ also has the property: $$\mathcal{W}_{c2} (t) = \mathcal{W}_{c1} (t) + (\mathcal{W}_{c2} (t) - *\mathcal{W}_{c1} (t))$$. Such an inheritance of the level sweeping property by the value function is specific to the case where the payoff function depends on two components of the phase vector. If it depends on three or more components of the vector x, the statement, generally speaking, is wrong. This is shown by a counterexample.

### Key words

Linear differential games value function level sets geometric difference complete sweeping

## Preview

### References

1. [1]
Botkin N.D., Evaluation of numerical construction error in differential game with fixed terminal time, Problems of Control and Information Theory, 11, 4, 283–295, 1982.
2. [2]
Chikrii A.A., Conflict controlled processes, Naukova dumka, Kiev, 1992 (in Russian).Google Scholar
3. [3]
Grigorenko N.L., Mathematical methods for control of a number of dynamic processes, Moscow State University, Moscow, 1990 (in Russian).Google Scholar
4. [4]
Gusjatnikov P.B. and Nikol’skii M.S., Pursuit time optimality, Doklady Akad. Nauk SSSR, 184,3, 518–521, 1969 (in Russian); Transl. as Sov. Math. Doklady, 10, 103–106, 1969.
5. [5]
Hadwiger H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin, 1957.
6. [6]
Isaacs R., Differential Games, John Wiley & Sons, New York, 1965.
7. [7]
Krasovskii N.N., Games Problems about Contact of Motions, Nauka, Moscow, 1970 (in Russian); Transl. as Rendez-vous Game Problems, Nat. Tech. Inf. Serv., Springfield, VA, 1971.Google Scholar
8. [8]
Krasovskii N.N. and Subbotin A.I., Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.
9. [9]
Kurzhanski A.B. and Valyi I., Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997.
10. [10]
Nikol’skii M.S., The First Direct Method of L.S. Pontryagin in Differential Games, Moscow State University, Moscow, 1984 (in Russian).Google Scholar
11. [11]
Polovinkin E.S., Ivanov G.E., Balashov M.V., Konstantiov R.V., and Khorev A.V., An algorithm for the numerical solution of linear differential games, Matematicheskii sbornik, 192,10, 95–122, 2001 (in Russian); Transl. as Sbornik: Mathematics, 192, 10, 1515–1542, 2001.Google Scholar
12. [12]
Ponomarev A.P. and Rozov N.H., Stability and convergence of alternating Pontryagin sums, Vestnik Moskov. Univ., Ser. XV Vyčisl. Mat. Kibernet., 1, 82–90, 1978 (in Russian).
13. [13]
Pontryagin L.S., On linear differential games, 1, Doklady Akad. Nauk SSSR, 174,6, 1278–1280, 1967 (in Russian); Transl. as. Sov. Math. Doklady, 8, 769–771, 1967.
14. [14]
Pontryagin L.S., On linear differential games, 2, Doklady Akad. Nauk SSSR, 175,4, 764–766, 1967 (in Russian); Transl. as Sov. Math. Doklady, 8, 910–912, 1967.Google Scholar
15. [15]
Pschenichnyi B.N. and Sagaidak M.I., Differential games of prescribed duration, Kibernetika, 2, 54–63, 1970 (in Russian); Transl. as Cybernetics, 6, 2, 72–80, 1970.Google Scholar
16. [16]
Rockafellar R.T., Convex Analysis, Princeton University Press, Princeton, NJ, 1970.