Structural discontinuities to approximate some optimization problems with a nonmonotone impulsive character

Bressan, Aldo ; Motta, Monica

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 6 (1995) no. 2, pp. 93-109

Résumé

In some preceding works we consider a class \( \mathcal{OP} \) of Boltz optimization problems for Lagrangian mechanical systems, where it is relevant a line \( l = l_{\gamma(\cdot)} \), regarded as determined by its (variable) curvature function \( \gamma(\cdot) \) of domain \( \left[ s_{0},s_{1} \right] \). Assume that the problem \( \widetilde{\mathcal{P}} \in \mathcal{OP} \) is regular but has an impulsive monotone character in the sense that near each of some points \( \delta_{1} \) to \( \delta_{\nu} \gamma(\cdot) \) is monotone and \( |\gamma'(\cdot)| \) is very large. In [10] we propose a procedure belonging to the theory of impulsive controls, in order to simplify \( \widetilde{\mathcal{P}} \) into a structurally discontinuous problem \( \mathcal{P} \). This is analogous to treating a biliard ball, disregarding its elasticity properties, as a rigid body bouncing according to a suitable restitution coefficient. Here the afore-mentioned treatment of \( \widetilde{\mathcal{P}} \) is extended to the case where its impulsive character fails to be monotone. Let \( c_{r,0} \) to \( c_{r,m_{r}} \) be the successive maxima and minima of \( \gamma(\cdot) \) or \( - \gamma(\cdot) \) near \( \delta_{r} (r = 1, \ldots, \nu) \). In constructing the problem \( \mathcal{P} \), which simplifies and approximates \( \widetilde{\mathcal{P}} \) as well as in [10] it is essential to approximate \( l_{\gamma(\cdot)} \) by means of a line \( l_{c(\cdot)} \) with \( c(\cdot) \) discontinuous only at \( \delta_{1}, \ldots, \delta_{\nu} \) and with \( |c'(\cdot)| \) never very large; furthermore now we must take the quantities \( c_{r,0} \) to \( c_{r,m_{r}} \) into account, e.g., by adding a «nonmonotonicity» type at \( \delta_{r} \), which vanishes in the monotone case \( (r = 1, \ldots, \nu) \). Starting from [10] we extend to the afore-mentioned general situation the notions of weak lower limit \( J^{*} \) of the functional to minimize, extended admissible process (which has an additional part in each \( \left[ c_{r,i-1},c_{r,i} \right] \)) and extended solution of the problem \( \mathcal{P} \), or better \( ( \mathcal{P}_{\nu}; \sigma_{r,1}, \ldots , \sigma_{r,m_{r}} ) \) where \( \sigma_{r,i} = c_{r,i} —c_{r,i-1} \)\( (i = 1, \ldots , m_{r}; r = 1, \ldots , \nu) \). In the general case we consider the extended (impulsive) original problem and the extended functional to minimize. This has an impulsive part at each of the points \( \delta_{1} \) to \( \delta_{\nu} \), as well as the differential constraints, complementary equations, and Pontrjagin's optimization conditions. Besides the end conditions at \( s_{0} \) and \( s_{1} \) there are junction conditions at \( \delta_{1} \) to \( \delta_{\nu} \). In the general case being considered we state a version of Pontrjagin's maximum principle and an existence theorem for the extended (impulsive) problem. We also study some properties of \( J^{*} \), e.g. when \( J^{*} \) is a weak minimum. In particular, within both the monotone case and the nonmonotone one, we show that the quantity \( J^{*} \), defined as a certain lower limit, equals the analogous limit; and this is practically a necessary and sufficient condition for the present approximation theory, started in [10], to be satisfactory.