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Hybrid dynamical systems, or hds: The ultimate switching experience
 In: Preprints of the Block Island Workshop on Control Using LogicBased Switching
, 1995
"... In previous work I have concentrated on formalizing the notion of a hybrid system as switching among an indexed collection of dynamical systems. I have also studied in some depth the modeling, analysis, and control of such systems. Here, I give a quick overview of the area of hybrid systems. I also ..."
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In previous work I have concentrated on formalizing the notion of a hybrid system as switching among an indexed collection of dynamical systems. I have also studied in some depth the modeling, analysis, and control of such systems. Here, I give a quick overview of the area of hybrid systems. I also briefly review the formal definition and discuss the main approaches taken in the study of hybrid systems. Finally, I elucidate issues in each of the research areas in light of previous results. 1
Biochemistry Department,
, 2003
"... proposed running head: probabilistic analysis of a differential equation for LP ..."
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proposed running head: probabilistic analysis of a differential equation for LP
Asa BenHur a,b, Joshua Feinberg c,d, Shmuel Fishman d,e
, 2001
"... In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the ..."
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In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find that the distribution of the convergence rate is a scaling function, namely it is a function of one variable that is a combination of three parameters: the number of variables, the number of constraints and the convergence rate, rather than a function of these parameters separately. We also estimate numerically the distribution of computation times, namely the time required to reach a vicinity of the attracting fixed point, and find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times.
Analog computation with dynamical systems
"... A b s t r a c t Physical systems exhibit various levels of complexity: their long term dynamics may converge to fixed points or exhibit complex chaotic behavior. This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are n ..."
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A b s t r a c t Physical systems exhibit various levels of complexity: their long term dynamics may converge to fixed points or exhibit complex chaotic behavior. This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes Pd, CoRPd, NPd and EXP,t, corresponding to their standard counterparts, according to the complexity of their long term behavior. The complexity of chaotic attractors relative to regular ones leads to the conjecture Pa:fi NPj. Continuous time flows have been proven useful in solving various practical problems. Our theory provides the tools for an algorithmic analysis of such flows. As an example we analyze the