Open-Loop Solvability Operator in Differential Games with Simple Motions in the Plane

Abstract

The paper deals with an open-loop solvability operator in two-person zero-sum differential games with simple motions. This operator takes a given terminal set to the set defined at the initial instant whence the first player can bring the control system to the terminal set if the player is informed about the open-loop control of the second player. It is known that the operator possesses the semigroup property in the case of a convex terminal set. In the paper, sufficient conditions ensuring the semigroup property in the non-convex case are formulated and proved for problems in the plane. Examples are constructed to illustrate the relevance of the formulated conditions. The connection with the Hopf formula is analysed.

Appendix

Let us consider a connection between the question investigated in the present work and the well-known Hopf formula (Alvarez et al. 1999; Bardi and Evans 1984; Hopf 1965). 1) For an arbitrary proper (i.e., not identically equal to + ∞) function $g: {\mathbb{R}}^{n} \rightarrow (-\infty,+\infty ]$, we define the Legendre transform

$$\displaystyle{{g}^{{\ast}}(s) =\sup _{x\in {\mathbb{R}}^{n}}[\langle x,s\rangle - g(x)],\quad s \in {\mathbb{R}}^{n}.}$$

By co g denote the convex hull of the function g. Properties of the function g ^∗ (Rockafellar 1970, Chap. 3, §16; Polovinkin and Balashov 2004) imply that if the proper function co g is continuous in ${\mathbb{R}}^{n}$, then

$$\displaystyle{ {(\mathrm{co}\,g)}^{{\ast}} = {g}^{{\ast}}. }$$

(12.36)

The support function ρ(⋅ , A) of a compact set $A \subset {\mathbb{R}}^{n}$ is connected with the indicator function

$$\displaystyle{\sigma _{A}(x) = \left \{\begin{array}{ll} 0, &\quad x \in A,\\ + \infty, &\quad x\not\in A \end{array} \right.}$$

of the set A by the relation

$$\displaystyle{ \rho (\cdot,A) =\sigma _{ A}^{{\ast}}(\cdot ). }$$

(12.37)

2) The Hopf formula

$$\displaystyle{ w(t,x):=\sup _{s\in {\mathbb{R}}^{n}}[\langle x,s\rangle {-\varphi }^{{\ast}}(s) + (\vartheta -t)H(s)],\ \ \ \ s \in {R}^{n},\ \ x \in {R}^{n},\ \ t \leq \vartheta }$$

(12.38)

represents the generalized (viscosity (Bardi and Capuzzo-Dolcetta 1997) or minimax (Subbotin 1991, 1995)) continuous solution of the Cauchy problem for the Hamilton–Jacobi equation

$$\displaystyle{ \begin{array}{rl} w_{t}(t,x) + H(w_{x}(t,x)) = 0,\quad &t \in (0,\vartheta ),\quad x \in {\mathbb{R}}^{n}; \\ w(\vartheta,x) =\varphi (x),\quad &x \in {\mathbb{R}}^{n},\end{array} }$$

(12.39)

with convex continuous terminal function $\varphi$ (Bardi and Evans 1984).

3) Set

$$\displaystyle{H(s) =\max _{q\in Q}\langle q,s\rangle +\min _{p\in P}\langle p,s\rangle,\quad \varphi (s) =\sigma _{M}(s),}$$

where M is a convex compact set. Consider the function $\overline{w}$ defined formally by the Hopf formula (12.38) for these data.

Let us show that

$$\displaystyle{ T_{\vartheta -t}(M) =\{ x \in {\mathbb{R}}^{n}:\ \overline{w}(t,x) \leq 0\}. }$$

(12.40)

As a preliminary, for the convex sets A and B, we establish the relation

$$\displaystyle{ \rho (\;\cdot \;,\;A\mathop{{-}}\limits^{{\ast}}B) =\mathrm{ co}\,{\bigl (\rho (\;\cdot \;,\;A) -\rho (\;\cdot \;,\;B)\bigr )}. }$$

(12.41)

We use the equality

$$\displaystyle{A\mathop{{-}}\limits^{{\ast}}B =\bigcap _{ b\in B}(A - b).}$$

It is known that the support function for the intersection of an arbitrary collection of compact convex sets coincides (Rockafellar 1970) with the convex hull of the function that is the minimum of support functions for the sets used in the intersection. In our case,

$$\displaystyle{\min _{b\in B}\rho (s,\;A - b) =\rho (s,\;A) +\min _{b\in B}\langle - b,\;s\rangle =\rho (s,\;A) -\rho (s,\;B).}$$

Consequently, (12.41) holds.

Applying (12.41), for the convex set M and $\tau =\vartheta -t$, we get

$$\displaystyle{ \rho (s,T_{\tau }(M)) =\mathrm{ co}\,\,h_{\tau }(s), }$$

(12.42)

where

$$\displaystyle{ h_{\tau }(s):=\rho (s,M) +\rho (s,-\tau P) -\rho (s,\tau Q) =\rho (s,M) -\tau H(s). }$$

(12.43)

The compactness of the sets P and Q implies that the function co h _τ is continuous. Using (12.36) and (12.42), we get

$$\displaystyle{ h_{\tau }^{{\ast}}(x) = {(\mathrm{co}\,\,h_{\tau })}^{{\ast}}(x) =\sup _{ s\in {\mathbb{R}}^{n}}[\langle x,s\rangle - (\mathrm{co}\,\,h_{\tau })(s)] =\sup _{s\in {\mathbb{R}}^{n}}[\langle x,s\rangle -\rho (s,T_{\tau }(M))]. }$$

(12.44)

On the other hand, by (12.43) and (12.37), we calculate

$$\displaystyle{ h_{\tau }^{{\ast}}(x) =\sup _{ s\in {\mathbb{R}}^{n}}[\langle x,s\rangle - h_{\tau }(s)] =\sup _{s\in {\mathbb{R}}^{n}}[\langle x,s\rangle -\sigma _{M}^{{\ast}}(s) +\tau H(s)]. }$$

(12.45)

Since

$$\displaystyle{T_{\vartheta -t}(M) =\{ x \in {\mathbb{R}}^{n}:\ \ \sup _{ s\in {\mathbb{R}}^{n}}[\langle x,s\rangle -\rho (s,T_{\vartheta -t}(M))] \leq 0\},}$$

applying (12.45), we write

$$\displaystyle{T_{\vartheta -t}(M) = \left \{x \in {\mathbb{R}}^{n}:\ \ h_{\tau }^{{\ast}}(x) \leq 0\right \}.}$$

Comparing (12.38), (12.44), and (12.45), we get (12.40).

4) Thus, if the compact set M is convex, then

$$\displaystyle{W_{0}(t) = T_{\vartheta -t}(M) =\{ x \in {\mathbb{R}}^{n}:\ \overline{w}(t,x) \leq 0\}.}$$

Now, in the case of a convex set M, we have two variants of useful description of the set W ₀(t), whence the guidance problem of the first player to the set M at the fixed instant $\vartheta$ is solvable, namely, by the Pshenichnyi formula and by the Hopf formula. The Pshenichnyi formula deals with sets, while the Hopf formula uses functions.

In the paper, for the problems in the plane, we obtain sufficient conditions to describe the set W ₀(t) by the Pshenichnyi formula for a non-convex set M. The Hopf formula does not work in the non-convex case.