Pursuit Evasion Differential Games in $$\boldsymbol{\ell^{p}}$$ on a Finite Time Interval

In this paper, we study linear optimal pursuit evasion games in Banach space of summable sequences with it pth power. We start by proving the existence and uniqueness of mild solutions as well as asymptotic stability for linear differential equations. Further, the dependence of asymptotic stability on the norm of the space is investigated: if the spectrum of the right-hand side lies strictly on the left half-plane then it is asymptotically (in fact, exponentially) stable for all spaces of summable sequences. We also show an example that is asymptotically stable for in the spaces of sequences, which is not asymptotically stable for in the space of essentially bounded sequences. We also study the linear differential game of pursuit and evasion and construct a strategy for the evader that guarantees the possibility of evasion from any initial state on a finite time interval and construct optimal strategy for the pursuer to complete the pursuit game.

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