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.

Abstract (VO)

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.

Riassunto

In precedenti lavori abbiamo considerato una classe \( \mathcal{OP} \) di problemi di ottimizzazione di Boltz per sistemi meccanici Lagrangiani, nei quali è rilevante una linea \( l = l_{\gamma(\cdot)} \), considerata come determinata dalla sua funzione (variabile) di curvatura \( \gamma(\cdot) \) di dominio \( \left[ s_{0},s_{1} \right] \). Il problema \( \widetilde{\mathcal{P}} \in \mathcal{OP} \) sia regolare ma abbia carattere impulsivo monotono nel senso che \( \gamma(\cdot) \) sia monotona e con \( |\gamma '(\cdot)| \) molto grande vicino a ciascuno di alcuni punti \( \delta_{1}, \ldots ,\delta_{\nu} \). In[10] abbiamo costruito un procedimento entro la teoria del controllo impulsivo, atto a semplificare \( \widetilde{\mathcal{P}} \) in un problema strutturalmente discontinuo \( \mathcal{P} \). Ciò è analogo al trattare una palla da bigliardo, anziché per es. con la teoria dell'elasticità, considerandola come un corpo rigido rimbalzante secondo un opportuno coefficiente di restituzione. Qui estendiamo la su accennata trattazione in [10] al caso che il carattere impulsivo di \( \widetilde{\mathcal{P}} \) sia non monotono. Siano \( c_{r,0}, \ldots , c_{r,m_{r}} \) i successivi massimi e minimi di \( \gamma(\cdot) \) o di \( - \gamma(\cdot) \) nella vicinanza di \( \delta_{r} (r = 1, \ldots, \nu) \). Nel costruire il problema \( \mathcal{P} \) semplificante e approssimante \( \widetilde{\mathcal{P}} \), come in [10] è ora essenziale considerare una linea \( l_{c(\cdot)} \) approssimante \( l_{\gamma(\cdot)} \) con \( c(\cdot) \) discontinua solo in \( \delta_{1}, \ldots, \delta_{\nu} \) e con \( |c'(\cdot)| \) mai molto grande; inoltre ora si deve tener conto delle suddette quantità \( c_{r,0} , \ldots , c_{r,m_{r}} \) per es., attraverso il «tipo di non monotonia» in \( \delta_{r} \), che svanisce nel caso monotono \( (r = 1, \ldots, \nu) \). Partendo da [10] estendiamo alla suddetta situazione generale le nozioni di estremo inferiore debole \( J^{*} \) del funzionale da minimizzare, processo ammissibile esteso (che ha parti addizionali in \( \left[ c_{r,i-1},c_{r,i} \right] \)) e soluzione estesa del problema \( \mathcal{P} \), o meglio \( ( \mathcal{P}_{\nu}; \sigma_{r,1}, \ldots , \sigma_{r,m_{r}} ) \) ove \( \sigma_{r,i} = c_{r,i} —c_{r,i-1} \)\( (i = 1, \ldots , m_{r}; r = 1, \ldots , \nu) \). Nel caso generale consideriamo pure il problema originale (impulsivo) esteso e il funzionale esteso da minimizzare. Questo ha parti impulsive nei punti \( \delta_{1} , \ldots , \delta_{\nu} \), al pari dei vincoli differenziali, delle equazioni complementari e delle condizioni di ottimizzazione di Pontrjagin. Oltre alle condizioni ai limiti in \( s_{0} \) ed \( s_{1} \) vi sono condizioni di giunzione in \( \delta_{1} , \ldots , \delta_{\nu} \). Nel detto caso generale enunciamo una versione del principio di massimo di Pontrjagin e un teorema di esistenza per il problema (impulsivo) esteso. Studiamo anche alcune proprietà di \( J^{*} \), tra l'altro quando esso è minimo debole. In particolare, nel caso monotono o no, mostriamo che la quantità \( J^{*} \), definita come un certo limite inferiore, eguaglia l'analogo limite; e ciò è praticamente una condizione necessaria e sufficiente affinché la presente teoria di approssimazione, iniziata in [10], sia soddisfacente.

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