Admissible Controllability for Linear Time-delay Systems in Banach Spaces : A problem in game theory
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We consider a linear control system with time-delays in a reflexive Banach space X : $dx(t)/dt=A_0x(t)+\int_(-n)^0 d\eta (s)x(t+s)+B(t)u(t) a.e. t>0.$ $x(0)=g^0, x(s)=g^1(s) a.e. s\in [-h,0],$ where $where (g^0,g^1)\in X\times L_p([-h,0] ; X), u\in L_q^(loc) (R^+ ; U),$ U is a reflexive Banach space, $p,q\in (1,\infty ),B(t)$ is a family of bounded linear operators on U to X and A$F_0$ generates a C$F_0$-semigroup, η is a Stieltjes measure. Moreover g$F^1$ and u are assumed to be restricted in ${g^1 ; \parallel g^1\parallel L_(p([-h,0]); X)) \leqq \rho $ and ${u ; \parallel u\parallel_ L_(p([0,T]);U)) \leqq \delta } (\rho \delta >0 ).$ For given $x^0,g^0\in X $ and a given time T>0, we discuss admissible controllability problems : (1) to determine independently g$F^1$ (･) and u (･) such that x(0)=g$F^0$ and $x(T)=x^0 or \parallel x(T)-x^0\parallel \leqq \epsilon $ (\epsilon : a given error). (the cooperative type) (2) to determine u (･) for a given g$F^1$ (･) such that x(0)=g$F^0$ and $\parallel x(T)-x^0\parallel \leqq \epsilon $(the noncooperative type) In this paper, for the both types, we establish necessary and sufficient conditions involving ρ and δ, in order that we can find such g$F^1$ and u. These conditions are expressed in terms of the fundamental solution of the homogeneous system with time-delays.
- The University of Tokyo,Department of Mathematics, Pusan National University|Department of Applied Mathematics, Faculty of Engineering, Kobe University|Department of Mathematics, College of Arts and Sciences, University of Tokyoの論文
The University of Tokyo,Department of Mathematics, Pusan National University|Department of Applied Mathematics, Faculty of Engineering, Kobe University|Department of Mathematics, College of Arts and Sciences, University of Tokyo | 論文
- Admissible Controllability for Linear Time-delay Systems in Banach Spaces : A problem in game theory