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...-linear or affine systems are considered, while the special case of autonomous systems, where the term u(t) is absent and the control variable consists solely of the switching times, is considered in =-=[11, 13, 23, 24]-=-. In particular, in [8,23,24] general nonlinear systems are considered together with nonlinear-programming algorithms that compute the gradient and second-order derivatives of the cost functional. Not...

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...ich the resulting hybrid Bellman equation guides the design of optimal control programs with, at most, N discrete transitions. 1 Introduction Optimal control of hybrid systems is certainly not a new topic. For example, the hybrid maximum principle has been well-studied [3,6], and the community now has a clear grasp of what constitutes necessary optimality conditions for very general classes of hybrid systems. Moreover, a number of results of a more computational flavor have complemented the work on the maximum principle, in which specialized classes of systems are considered. (See for example [1,4,7].) The contribution in this paper fits squarely in between the hybrid maximum principle work and the more computationally flavored work, in that we produce a Bellman equation for hybrid systems, along the lines of [2,5], that can be easily solved once a so-called optimality zone computation has been performed to seed the computation. 2 The Bimodal Hybrid System 2.1 Geometric Framework Given two open, connected, and simply connected regions D1, D2 such that D1 ∩D2 = ∅, forming a partition of the compact state space X in the sense that A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, L...

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...= f(t, s, u(t, s)), s ∈ [0, 1], t > 0 (1) ∂t ∂s for the unknown scalar function u(t, s). Assuming that a > 0, the material inflow at s = 0 shall be given by boundary data u(t, 0) = û(t; µ(t)), t ≥ 0, =-=(2)-=- where the inflow û is controlled by a parameter µ(t). The material distribution at t = 0 shall be given by initial data u(0, s) = ū(s), 0 < s < 1. (3) In R. Majumdar and P. Tabuada (Eds.): HSCC 2009,...

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...d in this way is useful in planning and control. Selection of subsequent control strategy, for example, may be dependent on terrain parameters. Similar derivations with respect to switching times [1]–=-=[3]-=- and parameters [4] independently can be found in previous work, however the derivations presented here are based on introductory calculus techniques, and the proof extends to cross-derivative terms i...

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...t. a cost function. The STO problem has been extensively studied for switched systems that consists of time continuous ordinary differential equations with switching laws, e.g. in [1], [2], [3], [4], =-=[5]-=-, [6], [7], [8] which we discuss in more detail in Section II. a) Problem setting: For simplicity, we consider a switched dynamical system on state space X ⊂ Rn that is described by only two different...

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...-linear or affine systems. The general nonlinear case of autonomous systems, where the term u(t) is absent and the control variable consists solely of the switching times, is considered in [8], [12], =-=[24]-=-, [25], [26]. In Ref. [8] we develop a simple formula for the partial derivatives dJ and subsequently propose a gradientdτi projection algorithm for computing optimal switching times. This paper deriv...

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...lves determining the optimal times for transitions between these modes by minimizing an objective function. Derivations of optimization algorithms for determining switching times can be found in [12]–=-=[14]-=-. Hybrid optimization is a particularly compelling framework for studying exploratory behavior for several reasons. At a high level, tactile exploration can be naturally presented as a hybrid system, ...

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...cation of [145], new interest has emerged in the area of optimal control computation. Recent trends in the area are: • The recognition that many practical problems involve impulsive or hybrid systems =-=[22, 126, 157]-=-; • The need to solve practical problems where the control takes values from a discrete set [46, 56, 111, 134]; • The practical need to determine a globally rather than just locally optimal solution [...