Martin von Mohrenschildt

Symbolic computation of Fenchel conjugates

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.

An Overview of a Formal Framework for Managing Mathematics

Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the IMPS Interactive Mathematical Proof System and the Axiom computer algebra system.

Hybrid systems: solutions, stability, control

This papers investigates the stability and control of hybrid systems. While stability is well defined and studied for continuous systems, hybrid systems (mixed continuous discrete systems) pose interesting and challenging new problems. The same holds for the question of control. We model hybrid systems using hybrid automata. The notion of observability and controllability is extended to hybrid system. We present a hybrid system adapted notion of stability including equilibrium and orbital stability, and show some implications. We discusses and define problems which arise in controlling hybrid systems, such as mode controllability.

Symbolic verification of hybrid systems: An algebraic approach

In this paper we present a new symbolic, computer algebra based approach to hybrid systems. Hybrid systems are systems containing both, continuous and discrete changing quantities. As is commonly done, we model hybrid systems using hybrid automata. Hybrid automata extend the classical notion of finite state machines by combining differential equations to model the dynamic behavior of systems with a finite control. In contrast to other approaches we consider hybrid automata as a generalization of differential equations and develop the notion of an “explicit symbolic solution” of a hybrid automaton. An explicit symbolic solution is an expression which gives the value of the quantities in question, the state variables, as a function of the design parameters and time. These solutions allow us to perform the verification of design properties. We present an algorithm leading to an implementation which computes these explicit symbolic solutions. We were able to detect design constraints on control systems that other methods fail to detect. This paper gives the basic definitions, algorithms, and three examples to demonstrate the advantage of the proposed approach.

Algebraic Composition of Function Tables

Abstract. In general, a program is composed of smaller program segments using composition, conditional constructs or loop constructs. We present a theory which enables us to algebraically define and compute the composition of conditional expressions. The conditional expressions are represented using tabular notation. The formal definition of the composition allows us to compute the close form representation of the composition of tabular expressions. The presented approach is based on a many sorted algebra containing information preserving composition. This formal definition of composition is then “lifted” to an extended algebra containing tabular expressions. The presented theory provides very compact algorithms and proofs.In general, a program is composed of smaller program segments using composition, conditional constructs or loop constructs. We present a theory which enables us to algebraically define and compute the composition of conditional expressions. The conditional expressions are represented using tabular notation. The formal definition of the composition allows us to compute the close form representation of the composition of tabular expressions. The presented approach is based on a many sorted algebra containing information preserving composition. This formal definition of composition is then “lifted” to an extended algebra containing tabular expressions. The presented theory provides very compact algorithms and proofs.

Simple type theory: simple steps towards a formal specification

Engineers, particularly software engineers, need to know how to read and write precise specifications. Specifications are made precise by expressing them in a formal mathematical language. Simple type theory, also as known as higher-order logic, is an excellent educational and practical tool for creating and understanding formal specifications. It provides a better logical foundation for specification than first-order logic and is a better introductory specification language than industrial specification languages like VDM-SL and Z. For these reasons, we recommend that simple type theory be incorporated into the undergraduate engineering curriculum.

PREDICTIVE TRACES IN HYBRID SYSTEMS

Hybrid systems are ideal models of real control systems where the discrete controller interacts with the continuous system on a closed loop via A/D converters and digitally controlled actuators. We extend the notion of a predictive controller to hybrid systems by introducing the notion of predictive traces, the set of all possible traces of a hybrid system starting in some state and mode into the future. A control algorithm is developed that explores this set of predictive traces to determine the next control action.