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...ss of hybrid systems considered in this contribution involves systems driven by continuous control inputs where switching is accompanied by a jump in the state. A similar class has been considered in =-=[33]-=-, where the authors focus attention on state delayed systems with controlled switches and where useful gradient formulas have been derived, for an application see [34]. See also [35] and [36] for rela...

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...f reading, we will use the alternate notation: Jɛ − J0 ∇J = lim ɛ→0 ɛ where J0 will be referred as the unperturbed system, and Jɛ as the perturbed system. Note this is quite similar to the program in =-=[8]-=-, [1]. First, we adjoin the dynamical constraints of Equations (1) and (2) into Equation (3) by defining Lagrange multipliers λi : [τi−1,τi] → R n and µi ∈ R n respectively. For the unperturbed system...

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...f reading, we will use the alternate notation: Jǫ − J0 ∇J = lim ǫ→0 ǫ where J0 will be referred as the unperturbed system, and Jǫ as the perturbed system. Note this is quite similar to the program in =-=[8]-=-, [1]. First, we adjoin the dynamical constraints of Equations (1) and (2) into Equation (3) by defining Lagrange multipliers λi : [τi−1, τi] → R n and µi ∈ R n respectively. For the unperturbed syste...

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...e of delays adds a nontrivial twist to the original problem posed in (Xu et al. 2002). We point out that the optimalsswitching problem bears some relation to the optimal impulsive control problem in (=-=Verriest et al. 2004-=-). Necessary conditions for the optimal switching policy are determined in Section 2. 2. VARIATIONAL APPROACH TO OPTIMAL SWITCHING Consider a system with a delay, τ. We follow standard notation (Hale ...