Control and Optimization with Differential-Algebraic Constraints

Abstract

Differential Algebraic Equations (DAEs) are mixed systems of differential and algebraic equations. It has been recognized for some time now that they have great potential both theoretically and in applications. DAEs form one of the most elegant and simple ways to model a physical system because they allow for the creation of separate models for subcomponents that can then be pasted together via a network. As a consequence, this concept is used in many modern CAD/modeling systems like SIMULINK, Scicos and DYMOLA, although most software packages cannot fully exploit the full potential of DAE models.But this nice feature of DAEs for modeling has also a disadvantage, since it shifts all of the difficulties of a system onto the analysis and the numerical methods. For this reason in recent years much effort has been spent to analyze general DAEs and to derive suitable numerical methods either for general DAEs, see e.g. [9,19,18,20,29,38] or for special DAEs arising in applications, see e.g. [6,12,28,37].This analytical and numerical work so far has been primarily driven by the simulation community, where the desire was to simulate the behavior of a complex system which could be electrical, mechanical, chemical, or all three. Due to the ever growing complexity of models which pose new challenges, the field is developing rather rapidly including now also hybrid [2,3,4,11,21,31,39] and delay systems [16,17,25,26,27].Once a system can be modeled and simulated, there arises the need to control the process or optimize its performance. The control of physical processes is an important task in many applications and over the last two decades there have been tremendous advances in the theory and applications of control in almost all disciplines of science and technologies. Note that this includes not only the obvious applications such as designing a more efficient process, but also determining what are the control mechanisms inherent in complex biological systems and fitting models to data. Recent Developments and Open ProblemsAs the two topics, simulation of DAEs and control/optimization, have evolved in recent years, there has been a growing awareness of interconnections which arise in a number of ways. One obvious way is the control of DAE modeled systems. Optimality conditions and a maximum principle for general DAEs have only recently been obtained [30] and the results are far from complete. But there are other more subtle connections in that the necessary conditions for an optimal control problem typically form a DAE and the solution of some control problems in the presence of constraints or invariants also involve DAEs. An open problem is in 1

Inexact Newton matrix-free methods for solving complex biotechnological systems

“…There have been a lot of methods for solving the nonlinear equations (5). The most popular and important are both the Newton and different variations of the inexact Newton methods [18].…”

“…The development of information technology, robust numerical methods and computing capacity, enables to obtain optimal operating policies of the complex biotechnological processes. An efficient solving of the differential-algebraic systems enables the use of optimization strategies, what can improve a process flow significantly [5], [10].…”

A Coupled System of Differential-Algebraic Equation and Hyperbolic Partial Differential Equation

“…However, due to their differences to elliptic and parabolic PDEs, hyperbolic PDEs are rarely discussed. Few books on optimal control with DAE constraints exist, for instance by Biegler, Campbell, and Mehrmann [18], but they do not include the optimal control of abstract DAEs.…”

“…18) by Lemma 2.3 where D + and E + are the linear operators induced by D + and E + respectively. Note that by Lemma 2.5 these are generalized inverses of D and E in the sense of Definition 2.4 but they are most certainly not generalized Moore-Penrose inverses; see the remark subsequent to this proof.…”