Optimal Control of Systems Determined by Strongly Nonlinear Operator Valued Measures
نویسنده
چکیده
In this paper we consider a class of distributed parameter systems (partial differential equations) determined by strongly nonlinear operator valued measures in the setting of the Gelfand triple V ↪→ H ↪→ V ∗ with continuous and dense embeddings where H is a separable Hilbert space and V is a reflexive Banach space with dual V ∗. The system is given by dx+A(dt, x) = f(t, x)γ(dt) +B(t)u(dt), x(0) = ξ, t ∈ I ≡ [0, T ] where A is a strongly nonlinear operator valued measure mapping Σ× V to V ∗ with Σ denoting the sigma algebra of subsets of the set I and f is a nonlinear operator mapping I ×H to H , γ is a countably additive bounded positive measure and the control u is a suitable vector measure. We present existence, uniqueness and regularity properties of weak solutions and then prove the existence of optimal controls (vector valued measures) for a class of control problems.
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