The purpose of this paper is twofold: (a) to provide a tutorial introduction to some key concepts from the theory of computational complexity, highlighting their relevance to systems and control theory, and (b) to survey the relatively recent research activity lying at the interface between these fields. We begin with a brief introduction to models of computation, the concepts of undecidability, polynomial time algorithms, NP-completeness, and the implications of intractability results. We then survey a number of problems that arise in systems and control theory, some of them classical, some of them related to current research. We discuss them from the point of view of computational complexity and also point out many open problems. In particular, we consider problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.

In this paper we describe and analyze models of problem solving and computation going beyond Turing Machines. Three principles of extending the Turing Machine's expressiveness are identified, namely, by interaction, evolution and infinity. Several models utilizing the above principles are presented. Other

Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+k , then any function of x computable by a Turing machine in T (x) time is also.

In this paper we show that Shannon’s General Purpose Analog Computer (GPAC) is equivalent to a particular class of recursive functions over the reals with the flavour of Kleene’s classical recursive function theory. We first consider the GPAC and several of its extensions to show that all these models have drawbacks and we introduce an alternative continuous-time model of computation that solve these problems. We also show that this new model preserve all the significant relations involving the previous models (namely, the equivalence with the differentially algebraic functions). We then continue with the topic of recursive functions over the reals, and we show full connections between functions generated by the model introduced so far and a particular class of recursive functions over the reals. 1

Continuous-valued recurrent neural networks can learn mechanisms for processing context-free languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown that qualitatively similar dynamics with similar constraints hold for a n b n c n , a context-sensitive language. The additional difficulty with a n b n c n , compared with the context-free language a n b n , consists of "counting up" and "counting down" letters simultaneously. The network solution is to oscillate in two principal dimensions, one for counting up and one for counting down. This study focuses on the dynamics employed by the Sequential Cascaded Network, in contrast with the Simple Recurrent Network, and the use of Backpropagation Through Time. Found solutions generalize well beyond training data, however, learning is not reliable. The contribution of this ...

this paper, a network of the former type will be analyzed in the following. Figure 24 shows a characteristic trajectory of a successful robot controller of this type. As above, the robot starts off facing the upper left obstacle. It turns away from it to the left, enters the zone, and collects three objects on its first pass through the zone, turning slightly to the left towards each of them. As soon as it has left the zone it starts moving in a semi-circle to the left, which takes it back into the zone. In the zone it starts moving straight ahead again, takes a slight turn to the right to collect the upper object, and continues straight ahead out of the zone. The same pattern is repeated: as soon as it leaves the zone, it moves in a semi-circle to the left, which takes it back into the zone, where it starts moving straight forward again. Once more it performs a slight turn to the right to collect an object it would otherwise have missed. It continues to move straight ahead, leaves the zone, returns in another semi-circle, enters once more and moves straight ahead until the evaluation period ends.

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This paper revisits one of the first models of analog computation, the General Purpose Analog Computer (GPAC). In particular, we restrict our attention to the improved model presented in [11] and we show that it can be further refined. With this we prove the following: (i) the previous model can be simplified; (ii) it admits extensions having close connections with the class of smooth continuous time dynamical systems. As a consequence, we conclude that some of these extensions achieve Turing universality. Finally, it is shown that if we introduce a new notion of computability for the GPAC, based on ideas from computable analysis, then one can compute transcendentally transcendental functions such as the Gamma function or Riemann’s Zeta function. 1

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