Optimal Control with Finite Control Set

Model Predictive Control (MPC) has gained huge popularity in the last decades and is nowadays considered as a main mathematical tool for industrial process control. The MPC method can be viewed as a method for constructing feedback solutions of long/infinite horizon Optimal Control Problems (OCPs), where the underlying controlled dynamics is described either in continuous or in discrete time. The feedback control, obtained by the method, has the advantage (as all feedback controls) to adapt to disturbances and measurement inaccuracies, but has the disadvantage (compared with other methods for feedback control design) that it has to be computed in real time. This leads to the necessity to utilize efficient computational methods, especially if the process to be controlled is fast, which requires high frequency of measurement and of resolving the OCPs involved. The project aims at developing certain theoretical and numerical aspects of the MPC framework, which are especially important in the case of fast processes. Such typically arise in power electronics the area that provides one of the main motivations for the project. Moreover, in this area the control inputs are implemented by switch on switch off devices, which, due to the finiteness of the feasible control set, brings combinatorial features into the problem. In addition, the OCPs involved are lacking the property called coercivity, which plays a very important role in both the theoretical analysis and the numerical solution of OCPs. The main goal of the project is to develop a new numerical approximation scheme that directly deals with non-coercive problems (without preliminary regularization) and directly produces switch on/off type controls (in contrast to the presently used methods). Such a method was recently proposed for restricted classes of problems by the author of the project with co- authors. However, the enhancement of this approximation approach and its embedding into the Newton-type iteration processes for differential variational inequalities that will be employed, requires to build a solid regularity theory for non-coercive OCPs, substantially extending the recently developed one. Apart from the MPC applications, the theoretical and computational tools created within the project will be useful for many other OCPs engineering and biomedicine, which are either non-coercive or such with finite control sets.

Model Predictive Control (MPC) is a main mathematical tool for industrial process control, digital engine control, and many other applications, including economics and epidemiology. It uses a mathematical model of a controlled dynamical system to generate approximately optimal control taking into account disturbances and measurement inaccuracies, being in the same time computationally efficient. This is of key importance, especially if the process to be controlled is fast, which requires high frequency of measurements and short computation time. Many practically important versions of MPC can be viewed as methods of online generation of reasonable (nearly optimal) feedback control under uncertainties. The mathematical analysis of such methods is based on certain regularity and stability properties of optimal open-loop and feedback controls. A number of basic mathematical results are established within the project, that facilitate this analysis. Conditions of regularity and stability of optimal controls are established for several classes of Optimal Control Problems (OCPs). Special attention is attributed to OCPs in which the optimal control takes values in a given finite set. The results are used for obtaining conditions for justification of versions of the MPC method and for estimation of the accuracy of the MPC generated control, depending on the size of the uncertainties. Several approaches to improving the efficiency of the numerical algorithms are developed and justified: the so-called "warm-staring", the utilization of high-order time-discretization schemes for the dynamics and the control, utilization of Newton-like methods. Related to the subject of the project, a new approach to solving OCPs on infinite horizon is developed, having numerous applications in mathematical economics. Moreover, with the intension of applying the MPC method, and in view of the COVID epidemic that emerged during the work on the project, a new model of controlled epidemic dynamics (especially appropriate for diseases with relatively long asymptomatic period) was developed and used for simulations and optimal policy analysis.