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Optimal impulse control for impulsive systems in Banach spaces. (English) Zbl 0959.49023

An impulse control system allows Dirac deltas (in time) in the control. The (more general) model used in this paper is a semilinear evolution equation \[ dy(t) = Ay(t) dt + f(y(t)) dt + g(y(t)) dv(t) , \quad y(0) = y_0 \tag{1} \] which holds in an interval \([0, T]\) minus a finite set of jumps \(\{t_1, t_2, \dots, t_n\} \subset (0, T).\) At the jumps, we have a condition \(y(t_k + 0) - y(t_k) = F_k(y(t_k)),\) where each \(F_k\) is a nonlinear operator. \(A\) is an infinitesimal generator and the function \(v(t)\) is of bounded variation in \(0 \leq t \leq T.\) The author discusses the theory of (1), sets up a general optimization problem of Bolza type for a suitable class of controls \(v(t),\) derives a necessary optimality condition and outlines an algorithm for computation of the optimal controls based on this condition.

MSC:

49N25 Impulsive optimal control problems
49K20 Optimality conditions for problems involving partial differential equations
49J27 Existence theories for problems in abstract spaces
49J20 Existence theories for optimal control problems involving partial differential equations
49K27 Optimality conditions for problems in abstract spaces
93C25 Control/observation systems in abstract spaces
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