49J20 Optimal control problems involving partial differential equations
Optimal Control Problems Governed by Nonlinear Partial Differential Equations and Inclusions
- The focus of this thesis lies on examining the solvability of optimal control problems constrained by nonlinear partial differential equations (PDE) and inclusions (PDI). There exist statements on the existence of solutions for optimal control problems with linear and semi-linear PDEs with monotone parts. The theory for non-monotone PDEs resp. the related optimal control problems is, to the author’s knowledge, incomplete regarding important issues. This concerns particularly the case of PDEs containing mappings, which only satisfy boundedness conditions on restricted sets. At first an optimal control problem is considered, which is characterized by a Laplace equation with Dirichlet boundary conditions and a nonlinear non-monotone Nemytskii operator. Under the decisive assumption of the existence of so called sub- and supersolutions for this differential equation and by introducing a truncation operator we can define an auxiliary problem which is characterized by a pseudomonotone operator. Thereby the solution theory for pseudomonotone operators of Brézis (1968) is applicable. Moreover, starting with the definitions of sub- und supersolution it can be shown, that every solution of the auxiliary problem is a solution of the original problem. The choice of a new optimal control problem which substitutes the original optimal control problem is again governed by the properties of the auxiliary operator. The equivalence of the auxiliary problem to the original problem and the existence of at least one solution can be shown. The technique of applying the Theorem of Lax-Milgram on a linearized problem can be adapted to the semi-linear non-monotone case. This procedure is already known from the theory of semi-linear monotone problems. For optimal control problems with quasi-linear differential equations, different methods are required. As in the semi-linear case, the property of pseudomonotonicity plays a key role in proving the existence of a solution of the quasi-linear PDE. In the proof of the existence of a solution for the optimal control problem other properties of the auxiliary operator are exploited. In the elliptic case operators which satisfy the S+ -property are important. In order to utilize this property, a transformation of the operator to some coercive auxiliary operator is necessary. For this reason a term is added, which penalizes the deviation from the admissible set of states. This term is characterized by a factor, which is derived explicitly in this work. The proof of the existence of a solution of the optimal control problem with parabolic equations is based on the definition of an auxiliary operator, coercivity and the S+ -property of operators. The set of solutions of the considered PDE is compact, but the number of solutions and the situation to each other is unknown. This leads to difficulties in deriving necessary optimality conditions. For this reason a direct approach to solve the optimal control problem with semi-linear PDEs is introduced. It is assumed, that the state constraints coincide with the sub- and the supersolution of the PDE with the upper and lower boundary of the control variable. Using an auxiliary operator, this assumption allows the formulation of an equivalent optimal control problem without pointwise state constraints. Through semi-discretization we can define a family of optimal control problems on a finite dimensional state-space. Existence of a subsequence of solutions of these optimal control problems which converges to a solution of the original problem is shown. Another important class of optimal control problems include differential inclusions which are described by multivalued operators. Quasi-linear elliptic inclusions are examined under global as well as local boundedness conditions. Under the assumption of global boundedness the properties of pseudomonotonicity and coercivity for a multivalued auxiliary operator are proven. The existence of at least one solution for the original inclusion follows from the application of a result from Hu and Papageorgiou (1997) on the auxiliary problem. The existence of at least one solution of the optimal control problem is proven by exploiting the coercivity of the multivalued auxiliary operator and the S+ -property of the non-multivalued part of this mapping. In the case of multivalued mappings of Clarke’s gradient type, the existence of at least one solution of the optimal control problem can be shown under local boundedness conditions. Elliptic as well as parabolic quasi-linear inclusions are considered. The proof is again based on coercivity and the S+ -property of the related auxiliary operators and the embedding properties of the spaces.
Shape Calculus Applied to Elliptic Optimal Control Problems
- This thesis is devoted to the analysis of a very simple, pointwisely state-constrained optimal control problem of an elliptic partial differential equation. The transfer of an idea from the field of optimal control of ordinary differential equations, which proved fruitful with respect to both theoretical treatment and design of algorithms, is the starting point. On this, the state inequality constraint, which is regarded as an equation inside the active set, is differentiated in order to obtain a control law.
A geometrical splitting of the constraints is necessary to carry over this approach to the chosen model problem. The associated assertions are rigorously ensured. The subsequent derivation of a control law in the sense of the abovementioned idea yields an equivalent reformulation of the model problem. The active set appears as an independent and equal optimization variable in this new formulation. Thereby a new class of optimization problem is established, which forms a hybrid of optimal control and shape-/topology optimization: set optimal control. This class is integrated into the very abstract framework of optimization on vector bundles; for that purpose some important notions from the field of calculus on manifolds are introduced and related with shape calculus.
First order necessary conditions of the set optimal control problem are derived by means of two different approaches: on the one hand a reduced approach via the elimination of the state variable, which uses a formulation as bilevel optimization problem, is pursued, and on the other hand a formal Lagrange principle is presented.
A comparison of the newly obtained optimality conditions with those known form literature yields relations between the Lagrange multipliers; in particular, it becomes apparent that the new approach involves higher regularity. The comparison is embedded to the theory of partial differential-algebraic equations, and it is shown that the new approach yields a reduction of the differential index.
Upon investigation of the gradient and the second covariant derivative of the objective functional different Newton- and trial algorithms are presented and discussed in detail. By means of a comparison with the well-established primal-dual active set method different benefits of the new approach become apparent. In particular, the new algorithms can be formulated in function space without any regularization. Some numerical tests illustrate that an efficient and competitive solution of state-constrained optimal control problems is achieved.
The whole work gives numerous references to different mathematical disciplines and encourages further investigations. All in all, it should be regarded as a first step towards a more comprehensive perspective on state-constrained optimal control of partial differential equations.