Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments

Optimal Switching for Hybrid Semilinear Evolutions

We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition times and Lagrange-type cost functions including switching costs. In particular, for a fixed sequence of modes, we derive necessary optimality conditions using an adjoint equation based representation for the gradient of the costs with respect to the switching times. For optimization with respect to the mode sequence, we discuss a mode-insertion gradient. The theory unifies and generalizes similar approaches for evolutions governed by ordinary and delay differential equations. More importantly, it also applies to systems governed by semilinear partial differential equations including switching the principle part. Examples from each of these system classes are discussed

State elimination for mixed-integer optimal control of partial differential equations by semigroup theory

Mixed-integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments

Penalty alternating direction methods for mixed-integer optimal control with combinatorial constraints

We consider mixed-integer optimal control problems with combinatorial constraints that couple over time such as minimum dwell times. We analyze a lifting and decom- position approach into a mixed-integer optimal control problem without combinatorial constraints and a mixed-integer problem for the combinatorial constraints in the control space. Both problems can be solved very efficiently with existing methods such as outer convexification with sum-up-rounding strategies and mixed-integer linear programming techniques. The coupling is handled using a penalty-approach. We provide an exactness result for the penalty which yields a solution approach that convergences to partial minima. We compare the quality of these dedicated points with those of other heuristics amongst an academic example and also for the optimization of electric transmission lines with switching of the network topology for flow reallocation in order to satisfy demands

Discrete optimal control with dynamic switches : outer approximation and branch-and-bound

Many real life applications lead to optimal control problems whose control is given in form of a finite set of switches. These switches can be operated within a given continuous time horizon and admit only a finite number of states. Examples include gear-switches in automotive engineering or valves and compressors in gas networks. Solving optimal control problems with discrete control variables is challenging, and this thesis aims at developing a branch-and-bound algorithm to globally solve such problems. We here focus on parabolic control problems with binary switches that have only finitely many switching points and possibly need to satisfy further combinatorial constraints. When no restrictions on the binary switches are considered, the straightforward continuous relaxations of the binary problems are closely related to the original problems, since any relaxed control can be approximated arbitrarily well by a sequence of binary switches using an increasing number of switchings. However, solving these relaxed problems and rounding the relaxed solution to produce a binary control, often fails when considering natural combinatorial switching constraints, such as, e.g. a minimum time span between two switchings of the same switch, or an upper bound on the total number of switchings. These constraints are typically treated in a heuristic postprocessing. In contrast, the combinatorial switching constraints are at the heart of our proposed branch-and-bound algorithm to globally solve the problems. The natural branching strategy, which fixes the value of the switches in finitely many points, combined with the bounded variation of the switches, guarantees that the non-fixed part of the switching pattern vanishes. Moreover, tight dual bounds are computed by completely describing the convex hull of feasible controls in function space. This description is built by cutting planes lifted from finite-dimensional projections of the set of feasible switches. The convexified problems can be solved by means of outer approximation. In this way, we compute safe dual bounds for the binary control problems, as long as we do not take the discretization error into account. To solve the problems in function space, we estimate the discretization error contained in the bounds. An adaptive refinement strategy is then specified to handle situations where the discretization-independent bound does not exclude that a solution of desired quality might exist in the current branch. Our branch-and-bound returns an ε -optimal solution in finite time for any given tolerance ε>0. Computational results illustrate the strength of our dual bounds and the potential of the proposed branch-and-bound algorithm