Optimal Control of Semilinear Parabolic Equations by BV-Functions

Here, we assume that Ω is a bounded domain in R, 1 ≤ n ≤ 3, with a Lipschitz boundary Γ, Q = Ω × (0, T ), Σ = Γ × (0, T ), and y0 ∈ L∞(Ω). BV (0, T ) denotes the space of bounded variation functions defined in (0, T ), with 0 < T < ∞ given. The controllers in (P) are supposed to be separable functions with respect to fixed spatial shape functions gj and free temporal amplitudes uj . The specific new feature in (P) is given by the choice of the control norm as the BVseminorm ‖uj‖M(0,T ). It enhances that the optimal controls are piecewise constant in time and that the number of jumps is penalized. The weights in (P) are assumed to satisfy αj > 0 and βj ≥ 0. Thus the goal of the optimal control problem (P) is to achieve a simple control strategy while simultaneously being as close to the target yd as possible. Let us further comment on the importance of this fact. If we consider the classical formulation of the control problem with a quadratic cost functional for the control, then the optimal control ū is equal to a multiple of the optimal adjoint state. Hence, while it is a regular function of time, its practical implementation can be involved in comparison to piecewise constant controls. Of course, ū can be approximated by piecewise constant functions, but a good approximation may require many jumps. Looking for a simpler structure for ū, one can consider the bang-bang formulation of the control problem by introducing pointwise constraints on the control: α ≤ u(t) ≤ β. Then, we can expect for ū to take only the values α and β. A drawback of this approach is given by the fact that ū frequently takes the extreme values all the time. This can lead to undesirable amounts of energy used to control the system. Our formulation pursues an optimal control ū with a simple structure and with lower energy than in the bang-bang case: We look for a piecewise constant control with just a few jumps. We show that this goal can be achieved with our formulation. The numerical tests also confirm the desired simple structure of the optimal controls.