Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

Abstract

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form $$\Vert u(t)\Vert _{L^1(\varOmega )} \le \gamma $$ ‖ u ( t ) ‖ L 1 ( Ω ) ≤ γ for $$t \in (0,T)$$ t ∈ ( 0 , T ) . This limits the total control that can be applied to the system at any instant of time. The $$L^1$$ L 1 -norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to $$\gamma $$ γ is investigated on the basis of different second order conditions.

Stability for Semilinear Parabolic Optimal Control Problems with Respect to Initial Data

“…The reader is referred to [5] and [6] for the proof of this result. Then, the mapping G : L r (0, T ; L s ( )) −→ L ∞ (Q) ∩ W (0, T ) given by G(u) = y u , the solution of (1.1), is well defined.…”

“…We refer to [5] for the proof of these theorems. Though the proof of Theorem 2.1 in [5] is performed for Dirichlet condition, the same arguments can be applied for the Neumann case with obvious modifications.…”

“…We refer to [5] for the proof of these theorems. Though the proof of Theorem 2.1 in [5] is performed for Dirichlet condition, the same arguments can be applied for the Neumann case with obvious modifications. In [5], the proof of Theorem 2.2 was carried out for s = 2, but it remains valid for our setting of r and s.…”

Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

“…Proof. Due to u \in L \infty (0, \infty ; L 2 (\omega )) and g \in L \infty (0, \infty ; L 2 (\Omega )), under the assumptions on f and inequality (2.5), the proof of existence and uniqueness of a solution y \in W (0, T )\cap L \infty (Q T ) for (2.1) for every T > 0 is standard; see, for instance, [8]. This proves the first statement of the theorem.…”

A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations

“…Moreover, as far as we know, the well posedness of the state equation (1.1) has not been studied for controls u ∈ L 2 (Q). In some recent papers, see [5,7,10], the existence of global minimizers to (P) in L ∞ (Q) has been proven in the absence of control constraints or for unbounded control sets with the restriction n ≤ 3 on the dimension.…”