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Measure solutions for impulsive systems in Banach spaces and their control. (English) Zbl 0951.34040
The notion of measure solutions to impulsive evolution equations in Banach spaces is introduced. This concept appears to be useful when the classical assumptions on the nonlinear functions in the system like Lipschitz continuity, linear growth, monotonicity, etc. fail to hold and, as a consequence, solutions in any of the classical senses (classical, strong, weak, or mild) may not exist. The author studies the existence (including regularity properties) of measure solutions to a semilinear impulsive evolution system in a Banach space and for a control system governed by a similar impulsive evolution system where the control has been introduced. These results are applied to control problems and the existence of optimal controls for a Bolza problem is proved.

MSC:
34G20 Nonlinear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
34H05 Control problems involving ordinary differential equations
34K35 Control problems for functional-differential equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
93C25 Control/observation systems in abstract spaces
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