Chapter 4 Optimal Control Problems in Infinite Dimensional Function

In this chapter, we will consider optimal control problems in function space where we will restrict ourselves to state equations that are both linear in the state and linear in the control: We assume that E is a reflexive, separable Banach space with norm ‖ · ‖E and dual space E∗ and X is another separable Banach space with norm ‖ · ‖X and separable dual space X∗. We further suppose A : D(A) ⊆ E → E∗ to be the infinitesimal generator of a compact, analytic semigroup S(·) in E (cf. section 4.2 below) and B : X∗ → E to be a bounded linear operator. Finally, let g : C([0, T ]; E) → C([0, T ]) and h : L∞([0, T ]; X∗) → L∞([0, T ]) be strictly convex, coercive and continuously Fréchet-differentiable mappings with Fréchet-derivatives g′(·) and h′(·), respectively.