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Controllability of evolution differential inclusions in Banach spaces. (English) Zbl 1128.93005

The authors study the controllability of distributed systems modeled by the evolution differential inclusion
\[ \frac{d}{dt}[y(t) - g(t,y(t))] \in A(t) y(t) + F(t,y(t)) + (Bu)(t), \; t \in [0,b].\tag{1} \]
In this equation, \(y(t) \in X\), \(u(t) \in U\), where \(X\) and \(U\) are Banach spaces, \(A(t)\) generates an evolution operator \(U(t,s)\) on \(X\), and \(F\) is a multivalued map. Assuming some appropriate boundedness conditions, and that the operator \(W: L^{2}([0,b],U) \to X\) defined by \[ W(u) = \int_{0}^{b} U(b,s) (Bu)(s) \, d s \] admits a bounded inverse modulo \(\ker W\), they establish that the system (1) is exactly controllable on \([0,b]\).

MSC:

93B05 Controllability
93C25 Control/observation systems in abstract spaces
34G25 Evolution inclusions
34H05 Control problems involving ordinary differential equations
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