Title: "Optimal control with non-renewable and reset-renewable resources."
Many realistic control problems involve multiple criteria for optimality and/or integral constraints on allowable controls. This can be conveniently modeled by introducing a budget for each secondary criterion/constraint. An augmented Hamilton-Jacobi-Bellman equation is then solved on an expanded state space, and its discontinuous viscosity solution yields the value function for the primary criterion/cost. The value function can be then used to recover all Pareto-optimal controls for all starting configurations.
For the case where the resources/budgets are monotone decreasing, we have developed a fast (non-iterative) algorithm based on an explicitly causal semi-Lagrangian discretization. The computational issues are similar to those encountered in "traditional" (single criterion) time-dependent optimal control, but with special (implicitly defined) state restrictions.
I will also address a more challenging case, where the resources can be instantaneously renewed (& budgets can be "reset") upon entering a pre-specified subset of the state space. This leads to a hybrid control problem with more subtle causal properties of the value function and additional challenges in constructing efficient numerical methods.
The first project is joint work with A.Kumar. The second project is joint work with R.Takei, W.Chen, Z.Clawson, and S.Kirov.
Alex Vladimirsky was born in Odessa (USSR) and his educational journey lead him from Moscow State University (1990) through UC San Diego
(1991-93) to UC Berkeley (B.A. in 1995; Ph.D. in 2001). He came to Cornell on an NSF Postdoctoral Fellowship in 2001 and has stayed there ever since then. Vladimirsky is a numerical analyst, whose research involves elements of Mathematics, Operations Research and Computer Science. As an analyst, he studies the effects of anisotropy and inhomogeneity on properties of differential equations and the computational efficiency of numerical methods. He works on a variety of discrete and continuous nonlinear problems that have some causal properties defining the direction of “information flow.” His applied interests include optimal control, differential games, robotics, seismic imaging, dimension reduction in chemical kinetics, approximation of invariant manifolds, pedestrian flow models, and homogenization.