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Optimal guaranteed control of linear systems under disturbances. (English. Russian original) Zbl 0956.49017
Math. Notes 60, No. 2, 147-152 (1996); translation from Mat. Zametki 60, No. 2, 198-205 (1996).
The author considers the following linear systems $\dot x(t)=A(t)x(t)-B(t)u(t)-D(t)w(t)+C(t)v(t), \tag{1}$ $$t\in [t^0,\theta], \;x(t^0)=x^0$$, where $$A(t), B(t), C(t), D(t)$$ are continuous matrices of dimensions $$n\times n$$, $$n\times m$$, $$n\times l$$, $$n\times k$$, respectively.
Parameters $$u$$ and $$w$$ are controls, and $$v$$ is a disturbance. It is assumed that $u(t)\in U(t), \tag{2}$ where $$t\mapsto U(t)\subset R_m$$ is a continuous convex- and compact-valued mapping.
The performance (cost) functional has the form $I=\gamma(x(\theta))+\int_\theta^{t^0} ( w_T (t)G(t)w(t)v_T (t)H(t)v(t)) dt, \tag{3}$ where $\;\gamma(x)=\min_{y\in M^0} (x-y)_T R^0 (x-y), \tag{4}$ $$G(t), H(t)$$ are symmetric positive semidefinite continuous matrices, $$R^0$$ is a positive definite matrix, $$M^0 \subset R_n$$ is the convex closed target set.
The aim of controls is minimizing the cost functional, the disturbance tries to maximize the cost functional.
It is proven that the differential game has the value function, i.e., there exists the equilibrium in the classes of feedbacks: Borel measurable controls $$u(t,x)$$ plus continuous controls $$w(t,x)$$ and continuous disturbances $$v(t,x)$$.
In the paper, simple formulas for calculating the value function and optimal guaranteed feedbacks are obtained.
##### MSC:
 49K35 Optimality conditions for minimax problems 49N10 Linear-quadratic optimal control problems 49N70 Differential games and control 93C73 Perturbations in control/observation systems 91A23 Differential games (aspects of game theory)
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