Abstract
This paper derives Hamilton-Jacobi equation (HJE) in Hilbert space for optimal control of stochastic distributed parameter systems (SDPSs) governed by partial differential equations (SPDEs) subject to both state-dependent and additive stochastic disturbances. First, nonlinear SDPSs are transformed to stochastic evolution systems (SESs), which are governed by stochastic ordinary differential equations (SODEs) in Hilbert space, using functional analysis. Second, the Hamilton-Jacobi equation (HJE), of which the solution results in an optimal control law, is derived. Third, a problem of optimal control of linear SDPSs, which include the air pollution process, with a quadratic cost functional is addressed as an application of the HJE. After, the control design is done, the SESs are transformed back to Euclidean space for implementation.
Highlights
Optimal control of DPSs, i.e., systems governed by partial differential equations (PDEs), has been under development since 1960s, optimal control of nonlinear stochastic distributed parameter systems (SDPSs) subject to stochastic disturbances has been rarely addressed, [6], [40], [39], [36], [27], [25], [5], [10], [23]
The first approach, referred to as the modal control one, discretizes the PDEs to obtain lumped-parameter systems described in terms of modal coordinates, i.e., systems of ordinary differential equations (ODEs), to which the classical control design methods [1], [21], [22] can be applied
After nonlinear SDPSs are transformed to stochastic evolution systems (SESs) in Hilbert space by using functional analysis, the goal of this paper is to present a derivation of the Hamilton-Jacobi equation (HJE) in Hilbert space, which is relatively easy to follow, for optimal control of SDPSs subject to both state-dependent and additive stochastic disturbances
Summary
Optimal control of DPSs, i.e., systems governed by partial differential equations (PDEs), has been under development since 1960s, optimal control of nonlinear SDPSs subject to stochastic disturbances has been rarely addressed, [6], [40], [39], [36], [27], [25], [5], [10], [23]. Several techniques proposed to solve the TPBV problem include [26], [31] on hill-climbing algorithms, and [20], [32], [17], [3], [4] on an extension of the representer method to nonlinear systems based on linearization. The HJE is applied to solve the optimal control problem of linear SDPSs, which include the air pollution process, with a quadratic cost functional
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