Mixed Frank–Wolfe Penalty Method with Applications to Nonconvex Optimal Control Problems

An Optimization Method for Semilinear Parabolic Relaxed Constrained Optimal Control Problems

“…From Step 3 of the Algorithm we have d l ≤ β l → 0. We use the above convergences and Lemma 3.3 to pass to the limit in (5.2), as l → ∞ and obtain Finally, we can show, see [5], that the constructed control r l k in Step 5 of the Algorithm can be chosen to be of Gamkrelidze type and these controls can be approximated by classical controls. So, the relaxed controls can be implemented.…”

“…Finally, we can show, see [5], that the constructed control r l k in Step 5 of the Algorithm can be chosen to be of Gamkrelidze type and these controls can be approximated by classical controls. So, the relaxed controls can be implemented.…”

“…For implementation reasons relaxed controls have to be approximated by classical ones. So, using standard techniques, the Gamkrelidze controls computed by the above method can be approximated by piecewise constant classical ones, see [5]. Thus the above method using relaxed controls has all the theoretical advantages of them and gives us at last, through the above-mentioned approximation, classical controls.…”

“…Relaxation had been introduced initially to prove the existence of optimal controls and then to derive necessary optimality conditions. Additionally, relaxed controls are used as a tool to develop optimization methods (Warga [4], Chryssoverghi et al [5]) and discrete approximation schemes (Chryssoverghi et al [6], Roubíček [7], Azhmyakov et al [8]). Relaxed controls have been applied to optimal control problems for systems defined by PDEs in [3], [2] as well as in many papers, among them [6], [9]- [13].…”

Discretization Methods for Nonconvex Optimal Control Problems with State Constraints

“…Therefore, the constructed control r nm k is of Gamkrelidze type for every k. Finally, discrete Gamkrelidze controls computed as above can then be approximated by piecewise constant classical controls using a standard procedure (see [2]). The control r nm k+1 yields then the same state, values of functionals, and G nm asr nm k .…”