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...by its N–th partial sum approximation via the Haar wavelet basis. To define this operation, FN : L2([0, 1],R)∩ BV ([0, 1],R)→ L2([0, 1],R) ∩BV ([0, 1],R), we employ the Haar wavelet (Section 7.2.2 in =-=[26]-=-): λ(t) = 1 if t ∈ [0, 12) , −1 if t ∈ [ 12 , 1) , 0 otherwise. (33) Letting 1 : R→ R be the constant function equal to one and bkj : [0, 1]→ R for k ∈ N and j ∈ {0, . . . , 2k−1}, be defined as...

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...us. Moreover, an ε–ball in the weak topology centered at ξ is denoted by: Nw(ξ, ε) = { ξ̄ ∈ Xr | ∥∥x(ξ) − x(ξ̄)∥∥ L2 < ε } . (29) A longer introduction to weak topology can be found in Section 3.8 in =-=[34]-=- or Section 2.3 in [24], but before continuing we make an important observation that aids in motivating the ensuing analysis. In order to understand the relationship between the topology generated by ...

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...mit the proof since it is a particular case of a well known result regarding the existence of unique minimizers of strictly convex functions over bounded sets in Hilbert spaces (Proposition II.1.2 in =-=[11]-=-). Theorem 4. For each ξ ∈ Xp, the map ξ′ 7→ ζ(ξ, ξ′), as defined in Equation (31), has a unique minimizer. 20 Employing these results we can prove the continuity of the optimality function. This resu...

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...1 dt. (5) We omit the proof of this result since it is beyond the scope of this paper. More details about the functions of bounded variation and weak derivatives can be found in Sections 3.5 and 9 in =-=[13]-=- and Section 5 in [43]. 2.2 Optimization Spaces We are interested in the control of systems whose trajectory is governed by a set of vector fields f : R×Rn× Rm ×Q → Rn, indexed by their last argument ...

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...f is Lipschitz with respect to all its arguments, the result follows as a direct extension of the classical existence and uniqueness theorem for nonlinear differential equations (see Section 2.4.1 in =-=[38]-=- for a standard version of this theorem). Therefore, since x(ξ) is unique, it is not an abuse of notation to denote the solution of Differential Equation (13) by x(ξ). Next, we develop an algorithm to...

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...eak derivative is the Dirac Delta, which is the weak derivative of the Step Function. The following result is fundamental in our analysis of functions of bounded variation: Theorem 1 (Exercise 5.1 in =-=[43]-=-). If f ∈ BV ([0, 1],Y), then f has a weak derivative, denoted ḟ . Moreover, ‖f‖BV = ∫ 1 0 ∥∥ḟ(t)∥∥ 1 dt. (5) We omit the proof of this result since it is beyond the scope of this paper. More detail...

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...minal work which presented many of the theoretical underpinnings of hybrid systems included a set of sufficient conditions for the optimal control of such systems using quasi variational inequalities =-=[5]-=-. Though compelling from a theoretical perspective, the application of this set of conditions to the construction of a numerical optimal control algorithm for hybrid dynamical systems requires the app...

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...N and t ∈ [0, 1]: (1) [FN (d)]i(t) ∈ [0, 1], (2) ∑q i=1[FN (d)]i(t) = 1, (3) [PN(FN (d))]i(t) ∈ {0, 1}, (4) ∑q i=1 [PN(FN (d))]i(t) = 1. Proof. Condition 1 follows due to the result in Section 3.3 in =-=[19]-=-. Condition 2 follows since the wavelet approximation is linear, thus, q∑ i=1 [FN (d)]i = q∑ i=1 〈di,1〉+ N∑ k=0 2k−1∑ j=0 〈di, bkj〉 bkj‖bkj‖2L2 = 〈1,1〉+ N∑ k=0 2k−1∑ j=0 〈1, bkj〉 bkj‖bkj‖2L2 = 1,...

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...es a running cost, then one can extend the existing state vector by introducing a new state, and modifying the cost function to evaluate this new state at the final time, as shown in Section 4.1.2 in =-=[29]-=-. By performing this type of modification, observe that each mode of the switched system can have a different running cost associated with it. Next, we define a family of functions, hj : Rn → R for j ...

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...lications including automobiles and locomotives employing different gears [20, 33], biological systems [15], situations where a control module has to switch its attention among a number of subsystems =-=[25, 32, 39]-=-, manufacturing systems [7] and situations where a control module has to collect data sequentially from a number of sensory sources [6, 9]. In addition, many complex nonlinear dynamical systems can be...

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...0 12 1.1 Related Work The algorithms to solve this switched system optimal control problem can be divided into two distinct groups according to whether they do or do not rely on the Maximum Principle =-=[28, 31, 37]-=-. Given the difficulty of the problem, both groups of approaches sometimes employ similar tactics during algorithm construction. A popular such tactic is one formalized by Xu et al. who proposed a bi-...

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...discrete modes of operation interact with continuous state evolution. Such systems have been used in a variety of modeling applications including automobiles and locomotives employing different gears =-=[20, 33]-=-, biological systems [15], situations where a control module has to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems [7] and situations where a control module has ...

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...son we compare the performance of Algorithm 2 on each of the examples to a traditional Mixed Integer Program (MIP). To perform this comparison, we employ a TOMLAB implementation of a MIP described in =-=[12]-=- which mixes branch and bound steps with sequential quadratic programming steps. Finally, all of our comparisons are performed on an Intel Xeon, 6 core, 3.47 GHz, 100 GB RAM machine. 56 Example Mode 1...

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...ed input. Though such a search is expensive, the existence of a discrete-valued input that approximates the behavior of the constructed relaxed discrete-valued input is proven by the Chattering Lemma =-=[4]-=-. Unfortunately this combinatorial search is unavoidable by employing the Chattering Lemma since it provides no means to construct a discrete-valued input that approximates a relaxed discrete-valued i...

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...locomotives employing different gears [20, 33], biological systems [15], situations where a control module has to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems =-=[7]-=- and situations where a control module has to collect data sequentially from a number of sensory sources [6, 9]. In addition, many complex nonlinear dynamical systems can be decomposed into simpler li...

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...as to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems [7] and situations where a control module has to collect data sequentially from a number of sensory sources =-=[6, 9]-=-. In addition, many complex nonlinear dynamical systems can be decomposed into simpler linear modes of operation that are more amenable to analysis and controller design [14, 16]. Given their utility,...

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... weak variations. Several have focused on the optimization of autonomous switched dynamical systems (i.e. systems without a continuous input) by fixing the mode sequence and working on devising first =-=[10]-=- and second order [22] numerical optimal control algorithms to optimize the amount of time spent in each mode. In order to extend these optimization techniques, Axelsson et al. employed the bi-level o...

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...lications including automobiles and locomotives employing different gears [20, 33], biological systems [15], situations where a control module has to switch its attention among a number of subsystems =-=[25, 32, 39]-=-, manufacturing systems [7] and situations where a control module has to collect data sequentially from a number of sensory sources [6, 9]. In addition, many complex nonlinear dynamical systems can be...

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...lications including automobiles and locomotives employing different gears [20, 33], biological systems [15], situations where a control module has to switch its attention among a number of subsystems =-=[25, 32, 39]-=-, manufacturing systems [7] and situations where a control module has to collect data sequentially from a number of sensory sources [6, 9]. In addition, many complex nonlinear dynamical systems can be...

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...Principle is the one proposed by Bengea et al. who relax the discrete-valued input and treat it as a continuous-valued input over which they can apply the Maximum Principle to perform optimal control =-=[3]-=-. A search through all possible discrete valued inputs is required in order to find one that approximates the trajectory of the switched system due to the application of the constructed relaxed discre...

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...who proposed a bi-level optimization scheme that at a low level optimized the continuous components of the problem while keeping the mode schedule fixed and at a high level modified the mode schedule =-=[42]-=-. We begin by describing the algorithms for switched system optimal control that rely on the Maximum Principle. One of the first such algorithms, presented by Alamir et al., applied the Maximum Princi...

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...t0 and tf , respectively. Second, we employ a MATLAB implementation of LSSOL from TOMLAB in order to compute the optimality function at each iteration of the algorithm since it is a quadratic program =-=[21]-=-. Third, for each example we employ a stopping criterion that terminates Algorithm 2, if θτ becomes too large. Each of these stopping criteria is described when we describe each example. Next, for the...

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...0 12 1.1 Related Work The algorithms to solve this switched system optimal control problem can be divided into two distinct groups according to whether they do or do not rely on the Maximum Principle =-=[28, 31, 37]-=-. Given the difficulty of the problem, both groups of approaches sometimes employ similar tactics during algorithm construction. A popular such tactic is one formalized by Xu et al. who proposed a bi-...

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...d to both finite-dimensional and infinite-dimensional optimization problems, the result is much weaker in the latter case. The issue mentioned above has been addressed several times in the literature =-=[2, 30, 40, 41]-=-, by formulating a stronger version of sufficient descent: Definition 5 (Definition 2.1 in [2]). Let S be a metric space, and consider the problem of minimizing the cost function J : S → R. A function...

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...rajectories when a forward pushing force is applied. The 3D dynamics of such needles has been described in [23] and the path planning in the presence of obstacles has been heuristically considered in =-=[8]-=-. The evolution of the needle can be defined using six dimensions where the first three dimensions represent the position of the needle relative to the point of entry and the last three dimensions rep...

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...mization scheme proposed by Xu et al. and applied the Maximum Principle to perform optimization at the lower level and applied the Hamming distance to compare different possible nearby mode schedules =-=[35]-=-. Given the algorithm that we construct in this paper, the most relevant of the approaches that rely on the Maximum Principle is the one proposed by Bengea et al. who relax the discrete-valued input a...

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...ip Flexible Needle Bevel-tip flexible needles are asymmetric needles that move along curved trajectories when a forward pushing force is applied. The 3D dynamics of such needles has been described in =-=[23]-=- and the path planning in the presence of obstacles has been heuristically considered in [8]. The evolution of the needle can be defined using six dimensions where the first three dimensions represent...

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...rming several single mode insertions at each iteration [40]. Second, Gonzalez et al., extend the approach to make it applicable to constrained switched dynamical systems with a continuousvalued input =-=[17, 18]-=-. Though these single mode insertion techniques avoid the computational expense of considering all possible mode schedules during the high-level optimization, this improvement comes at the expense of ...

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...as to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems [7] and situations where a control module has to collect data sequentially from a number of sensory sources =-=[6, 9]-=-. In addition, many complex nonlinear dynamical systems can be decomposed into simpler linear modes of operation that are more amenable to analysis and controller design [14, 16]. Given their utility,...

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...mber of sensory sources [6, 9]. In addition, many complex nonlinear dynamical systems can be decomposed into simpler linear modes of operation that are more amenable to analysis and controller design =-=[14, 16]-=-. Given their utility, there has been considerable interest in devising algorithms to perform optimal control of such systems. In fact, even Branicky et al.’s seminal work which presented many of the ...

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...m optimal control that rely on the Maximum Principle. One of the first such algorithms, presented by Alamir et al., applied the Maximum Principle directly to a discrete time switched dynamical system =-=[1]-=-. In order to construct such an algorithm for a continuous time switched dynamical system, Shaikh et al. employed the bi-level optimization scheme proposed by Xu et al. and applied the Maximum Princip...

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...ral have focused on the optimization of autonomous switched dynamical systems (i.e. systems without a continuous input) by fixing the mode sequence and working on devising first [10] and second order =-=[22]-=- numerical optimal control algorithms to optimize the amount of time spent in each mode. In order to extend these optimization techniques, Axelsson et al. employed the bi-level optimization strategy p...

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... q = 2), but as the author’s of the theorem remark, their proof can be generalized to an arbitrary number of vector fields. A particular version of this existence theorem can also be found in Lemma 1 =-=[36]-=-. Theorem 3 says that the behavior of any element of the relaxed optimization space with respect to the trajectory of switched system can be approximated arbitrarily well by a point in the pure optimi...

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...nteract with continuous state evolution. Such systems have been used in a variety of modeling applications including automobiles and locomotives employing different gears [20, 33], biological systems =-=[15]-=-, situations where a control module has to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems [7] and situations where a control module has to collect data sequentia...

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...e insertion technique [2]. There have been two major extensions to Axelsson et al.’s algorithm. First, Wardi et al., extend the approach by performing several single mode insertions at each iteration =-=[40]-=-. Second, Gonzalez et al., extend the approach to make it applicable to constrained switched dynamical systems with a continuousvalued input [17, 18]. Though these single mode insertion techniques avo...

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...rming several single mode insertions at each iteration [40]. Second, Gonzalez et al., extend the approach to make it applicable to constrained switched dynamical systems with a continuousvalued input =-=[17, 18]-=-. Though these single mode insertion techniques avoid the computational expense of considering all possible mode schedules during the high-level optimization, this improvement comes at the expense of ...

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...y their numerical implementation for nonlinear switched systems is fundamentally restricted due to their reliance on approximating strong or needle variations with arbitrary precision as explained in =-=[27]-=-. Next, we describe the algorithms that do not rely on the Maximum Principle but rather employ weak variations. Several have focused on the optimization of autonomous switched dynamical systems (i.e. ...

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...d to both finite-dimensional and infinite-dimensional optimization problems, the result is much weaker in the latter case. The issue mentioned above has been addressed several times in the literature =-=[2, 30, 40, 41]-=-, by formulating a stronger version of sufficient descent: Definition 5 (Definition 2.1 in [2]). Let S be a metric space, and consider the problem of minimizing the cost function J : S → R. A function...

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...discrete modes of operation interact with continuous state evolution. Such systems have been used in a variety of modeling applications including automobiles and locomotives employing different gears =-=[20, 33]-=-, biological systems [15], situations where a control module has to switch its attention among a number of subsystems [25, 32, 39], manufacturing systems [7] and situations where a control module has ...

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...d to both finite-dimensional and infinite-dimensional optimization problems, the result is much weaker in the latter case. The issue mentioned above has been addressed several times in the literature =-=[2, 30, 40, 41]-=-, by formulating a stronger version of sufficient descent: Definition 5 (Definition 2.1 in [2]). Let S be a metric space, and consider the problem of minimizing the cost function J : S → R. A function...