Advances in Mathematical Modeling, Optimization and Optimal Control

This book constitutes a collection of developed versions of plenary papers presented (with one exception) at the 16th French–German–Polish Conference on Optimization, held in Kraków in 2013. They are authored by researchers of international repute in the field of optimization and optimal control. The book includes a number of new theoretical results and applications in biomechanics, medical technology, image processing, robot control, etc.

The aim of this paper is to provide an overview of recent development related to Bregman distances outside its native areas of optimization and statistics. We discuss approaches in inverse problems and image processing based on Bregman distances, which have evolved to a standard tool in these fields in the last decade. Moreover, we discuss related issues in the analysis and numerical analysis of nonlinear partial differential equations with a variational structure. For such problems Bregman distances appear to be of similar importance, but are currently used only in a quite hidden fashion. We try to work out explicitly the aspects related to Bregman distances, which also lead to novel mathematical questions and may also stimulate further research in these areas.

In this article we study the operator version of a first order in time partial differential inclusion as well as its time discretization obtained by an implicit Euler scheme. This technique, known as the Rothe method, yields the semidiscrete trajectories that are proved to converge to the solution of the original problem. While both the time continuous problem and its semidiscretization can have nonunique solutions we prove that, as times goes to infinity, all trajectories are attracted towards certain compact and invariant sets, so-called global attractors. We prove that the semidiscrete attractors converge upper-semicontinuously to the global attractor of time continuous problem.

A class of nonsmooth shape optimization problems for variational inequalities is considered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith’s functional, which is defined in plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard directional differentiability of metric the projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. A domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis. The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. Singular geometrical domain perturbations in an elastic body \(\Omega\) are approximated by regular perturbations of bilinear forms in variational inequality, without any loss of precision for the purposes of the second-order shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations \(\epsilon \rightarrow \omega _{\epsilon }\subset \Omega\) centred at \(\hat{x} \in \Omega\) are replaced by regular perturbations of bilinear forms supported on the manifold \(\Gamma _{R} =\{ \vert x -\hat{ x}\vert = R\}\) in an elastic body, with R > ε > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems.

This paper gives an overview of the mathematical background and possible applications of optimal control and inverse optimal control in the field of medical and rehabilitation technology, in particular in human movement analysis, therapy and improvement by means of appropriate medical devices. One particular challenge in this area is the formulation of suitable subject-specific models of motions for healthy and impaired humans including skeletal multibody dynamics and potentially neuromuscular components, and their combination with models of the technical components. The formulation of hybrid multi-phase optimal control problems arising in this context involves non-standard elements such as the open- or closed loop stability of the dynamic motion. Efficient methods for the solution of optimal control and inverse optimal control are discussed and particular difficulties of this problem class are highlighted. In addition, we present several example applications of these methods in the development of mobility aids for geriatric patients, the optimization-based design of exoskeletons, the analysis of running motions with prostheses, the optimal functional electrical stimulation of hemiplegic patients, as well as stability studies for different types of movement.

We survey the results on no-gap second-order optimality conditions (both necessary and sufficient) in the Calculus of Variations and Optimal Control, that were obtained in the monographs Milyutin and Osmolovskii (Calculus of Variations and Optimal Control. Translations of Mathematical Monographs. American Mathematical Society, Providence, 1998) and Osmolovskii and Maurer (Applications to Regular and Bang-Bang Control: Second-Order Necessary and Sufficient Optimality Conditions in Calculus of Variations and Optimal Control. SIAM Series Design and Control, vol. DC 24. SIAM Publications, Philadelphia, 2012), and discuss their further development. First, we formulate such conditions for broken extremals in the simplest problem of the Calculus of Variations and then, we consider them for discontinuous controls in optimal control problems with endpoint and mixed state-control constraints, considered on a variable time interval. Further, we discuss such conditions for bang-bang controls in optimal control problems, where the control appears linearly in the Pontryagin-Hamilton function with control constraints given in the form of a convex polyhedron. Bang-bang controls induce an optimization problem with respect to the switching times of the control, the so-called Induced Optimization Problem. We show that second-order sufficient condition for the Induced Optimization Problem together with the so-called strict bang-bang property ensures second-order sufficient conditions for the bang-bang control problem. Finally, we discuss optimal control problems with mixed control-state constraints and control appearing linearly. Taking the mixed constraint as a new control variable we convert such problems to bang-bang control problems. The numerical verification of second-order conditions is illustrated on three examples.