Dynamic systems with a mix of continuous and discrete components, often called hybrid dynamic systems, frequently arise in engineering applications. Since many of these applications are safety critical, it is important to use reliable methods to simulate hybrid systems.This paper illustrates two approaches to rigorous simulation of hybrid dynamic systems. In the first approach, we use symbolic methods to compute closed-form solutions, thus avoiding round off and truncation errors. In the second approach, we use interval methods to compute rigorous bounds on the solution of a hybrid system.
Convex optimization deals with certain classes of mathematical optimization problems including least-squares and linear programming problems. This area has recently been the focus of considerable study and interest due to the facts that convex optimization problems can be solved efficiently by interior-point methods and that convex optimization problems are actually much more prevalent in practice that previously thought.Key notions in convex optimization are the Fenchel conjugate and the subdifferential of a convex function. In this paper, we build a new bridge between convex optimization and symbolic mathematics by describing the Maple package fenchel, which allows for the symbolic computation of these objects for numerous convex functions defined on the real line. We are able to symbolically reproduce computations for finding Fenchel conjugates and subdifferentials for numerous nontrivial examples found in the literature.
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