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ON THE CONTINUOUS SELECTIONS OF SOLUTION SETS OF LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL INCLUSIONS

Abdus Salam

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

We prove that a multifunction associated with the set of solutions of Lipschitzian quantum stochastic dieren tial inclusion (QSDI) admits a selection continuous from some subsets of complex numbers to the space of the matrix elements of adapted weakly absolutely continuous quantum stochastic processes. In particular, we show that the solution set map as well as the reachable set of the QSDI admit some continuous representations.

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We prove that a multifunction associated with the set of solutions of Lipschitzian quantum stochastic dieren tial inclusion (QSDI) admits a selection continuous from some subsets of complex numbers to the space of the matrix elements of adapted weakly absolutely continuous quantum stochastic processes. In particular, we show that the solution set map as well as the reachable set of the QSDI admit some continuous representations.
ABSTRACT. We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection which is continuous from the locally convex space of stochastic processes to the space of adapted and weakly absolutely continuous solutions. As a corollary, the reachable set multifunction admits a continuous selection. In the framework of the Hudson-Parthasarathy formulation of quantum stochastic calculus, these results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion.
We prove that the map that associates to the initial value the set of solutions to the Lipschitzian Quantum Stochastic Differential Inclusion (QSDI) admits a selection continuous from the locally convex space of stochastic processes to the adapted and weakly absolutely continuous space of solutions. As a corollary, we show that the reachable sets admit some continuous selections. In the framework of the Hudson - Parthasarathy formulations of quantum stochastic calculus, our results are achieved subject to some compactness conditions on the set of initial values and on some coefficients of the inclusion. The results here compliment similar results in our previous work in [3] where continuous selections defined on the set of the matrix elements of initial values were established. JONAMP Vol. 11 2007: pp. 71-82
Given any finite set of trajectories of a Lipschitzian quantum stochastic differential inclu-sion (QSDI), there exists a continuous selection from the complex-valued multifunction associated with the solution set of the inclusion, interpolating the matrix elements of the given trajectories. Furthermore, the difference of any two of such solutions is bounded in the seminorm of the locally convex space of solutions. Copyright © 2007 E. O. Ayoola and J. O. Adeyeye. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] Wolenski, P.R. 1990. The exponential formula for the reachable set of a Lipschitz differential inclusion. SIAM J. Control Optim., 28(5): 1148–1161. [Crossref], [Web of Science ®] , [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.