The inextricable interplay between the dual problems of optimal control and estimation forms the basis for effective decision theory in successful applications of science and engineering. The development of feedback control laws is a central problem in the optimal control theory. Since feedback control utilize the current state information, as opposed to the state information from an earlier epoch, the resulting closed loop system is modestly robust to model errors and state uncertainties. The problem of finding optimal feedback control laws for nonlinear dynamical systems remains a significant challenge, even in the case when perfect state information is available. This is because, the feedback solution is typically derived from the so-called value function, which is the solution to a nonlinear partial differential equation known as the Hamilton Jacobi Bellman (HJB) equation. On the other hand, the evolution of the Probability Density Function (PDF) associated with the state is governed by the Fokker-Planck-Kolmogorov (FPK) equation. When the system dynamics is governed by linear differential equations, the state uncertainties are modeled by a Gaussian process, and the control objective is to minimize a quadratic cost functional that is tantamount to the energy of the dynamical system, it becomes possible to analytically solve these Partial Differential Equations (PDEs). The solution process in this case leads to the celebrated Linear Quadratic Gaussian (LQG) feedback control, that leverages the certainty equivalence principle to provide an estimated state feedback control law, with the optimal state estimates obtained from the Kalman Filter. However, in the presence of system nonlinearities, Kalman’s separation theorem ceases to hold and the analytical solution of the PDEs cannot be obtained.
The chief impediment in the task of envisioning a systematic solution process for these PDEs is associated with the fact that the number of spatial variables is equal to the state dimension, which is twice the number of degrees of freedom of a mechanical system. For instance, the problem of stabilization of a rigid body is a 12 dimensional problem, and simple finite element discretization strategies entail a large number of variables for effective solution process. Bellman famously termed this explosive growth as the curse of dimensionality. Furthermore, unique challenges exist in the pursuit of solution methodologies for each PDE. Difficulties associated with the solution of the HJB equation can be attributed to the nonlinearity of the dynamical system. The process is compounded by the presence of a specified terminal manifold at which we intend the state to reach at a fixed final time, where the dimensionality is increased further, to invoke mechanisms to make the feedback control process aware of the terminal manifold to be attained at the fixed terminal time. The challenge in solving the FPK PDE lies in the determination of the domain of the state PDF at any given instant of time. It is typically impossible to make this determination apriori, and one has to resolve this causality dilemma.
The main focus of the proposed work is to develop a computationally efficient unified approach to solve linear and nonlinear PDEs with several independent variables. The crux of this research effort is to lay foundation for a computationally tractable unified framework to synthesize optimal feedback controllers and design finite-dimensional high fidelity nonlinear filters for stochastic nonlinear systems. The approach break the curse of dimensionality by employing a unique combination of non-product quadrature methods and sparse approximation tools. A salient feature of the proposed approach is its non-parametric nature, where the solution process does not assume any structure for the field variable of interest.