Continuous Selections of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

A multifunction associated with the set of solutions of Lipschitzian quantum stochastic differential 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, the solution set map as well as the reachable set of the QSDI admit some continuous representations.     

Topological Properties of Solution Sets of Lipschitzian Quantum Stochastic Differential Inclusions

We establish a continuous mapping of the space of the matrix elements of an arbitrary nonempty set of quasi solutions of Lipschitzian quantum stochastic differential inclusion (QSDI) into the space of the matrix elements of its solutions. As a corollary, we furnish a generalization of a previous selection result. In particular, when the coefficients of the inclusion are integrably bounded, we show that the space of the matrix elements of solutions is an absolute retract, contractible, locally and integrally connected in an arbitrary dimension. Permanent address: E.O. Ayoola, Department of Mathematics, University of Ibadan, Ibadan, Nigeria.     

CONSTRUCTION OF APPROXIMATE ATTAINABILITY SETS FOR LIPSCHITZIAN QUANTUM STOCHASTIC DIFFERENTIAL INCLUSIONS

We present a numerical method for constructing, with a specified accuracy attainability sets for Lipschitzian quantum stochastic differential inclusions. Results here generalize the Komarov-Pevchikh results concerning classical differential inclusions to the present noncommutative quantum setting involving unbounded linear operators on a Hilbert space.

AMS Subject Classification (1991): 60H10, 60H20, 65L05, 81S25.     

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

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.  [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.     

Error Estimates for Discretized Quantum Stochastic Differential Inclusions

Abstract

This paper is concerned with the error estimates involved in the solution of a discrete approximation of a quantum stochastic differential inclusion (QSDI). Our main results rely on certain properties of the averaged modulus of continuity for multivalued sesquilinear forms associated with QSDI. We obtained results concerning the estimates of the Hausdorff distance between the set of solutions of the QSDI and the set of solutions of its discrete approximation. This extends the results of Dontchev and Farkhi Dontchev, A.L.; Farkhi, E.M. (Error estimates for discre‐ tized differential inclusions. Computing 1989, 41, 349–358) concerning classical differential inclusions to the present noncommutative quantum setting involving inclusions in certain locally convex space.