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Strongly nonlinear impulsive evolution equations and optimal control. (English) Zbl 1065.49023
The system is described by the evolution equation \[ x'(t) + A(t, x(t)) = g(t, x(t)) + B(t)u(t) \] satisfied in an interval \(0 < t < T\) minus a finite set \(\{t_k\},\) \(0 < t_1 < t_2 < \dots < t_n < T.\) The solution satisfies an initial condition at \(t = 0\) and jump conditions at the other points, \[ x(0) = x_0, \quad x(t_k^+) - x(t_k) = F_k(x(t_k)) , \quad k = 1, 2, \dots, n . \] The author discusses existence of solutions to a control problem that consists of minimizing an integral functional \(J(x, u)\) over trajectories of the system.

MSC:
49N25 Impulsive optimal control problems
49J27 Existence theories for problems in abstract spaces
34G20 Nonlinear differential equations in abstract spaces
93C25 Control/observation systems in abstract spaces
34A37 Ordinary differential equations with impulses
49J15 Existence theories for optimal control problems involving ordinary differential equations
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