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Register a new userIndefinite Backward Stochastic Linear-Quadratic Optimal Control Problems
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This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are present in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished under a fairly weak condition by investigating the connection with forward stochastic LQ optimal control problems.
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This paper applies a reinforcement learning (RL) method to solve infinite horizon continuous-time stochastic linear quadratic problems, where drift and diffusion terms in the dynamics may depend on both the state and control. Based on Bellmans dynamic programming principle, an online RL algorithm is presented to attain the optimal control with just partial system information. This algorithm directly computes the optimal control rather than estimating the system coefficients and solving the related Riccati equation. It just requires local trajectory information, greatly simplifying the calculation processing. Two numerical examples are carried out to shed light on our theoretical findings.
We are concerned with the linear-quadratic optimal stochastic control problem with random coefficients. Under suitable conditions, we prove that the value field $V(t,x,omega), (t,x,omega)in [0,T]times R^ntimes Omega$, is quadratic in $x$, and has the following form: $V(t,x)=langle K_tx, xrangle$ where $K$ is an essentially bounded nonnegative symmetric matrix-valued adapted processes. Using the dynamic programming principle (DPP), we prove that $K$ is a continuous semi-martingale of the form $$K_t=K_0+int_0^t , dk_s+sum_{i=1}^dint_0^tL_s^i, dW_s^i, quad tin [0,T]$$ with $k$ being a continuous process of bounded variation and $$Eleft[left(int_0^T|L_s|^2, dsright)^pright] <infty, quad forall pge 2; $$ and that $(K, L)$ with $L:=(L^1, cdots, L^d)$ is a solution to the associated backward stochastic Riccati equation (BSRE), whose generator is highly nonlinear in the unknown pair of processes. The uniqueness is also proved via a localized completion of squares in a self-contained manner for a general BSRE. The existence and uniqueness of adapted solution to a general BSRE was initially proposed by the French mathematician J. M. Bismut (1976, 1978). It had been solved by the author (2003) via the stochastic maximum principle with a viewpoint of stochastic flow for the associated stochastic Hamiltonian system. The present paper is its companion, and gives the {it second but more comprehensive} adapted solution to a general BSRE via the DDP. Further extensions to the jump-diffusion control system and to the general nonlinear control system are possible.
The linear-quadratic regulator (LQR) is an efficient control method for linear and linearized systems. Typically, LQR is implemented in minimal coordinates (also called generalized or joint coordinates). However, other coordinates are possible and recent research suggests that there may be numerical and control-theoretic advantages when using higher-dimensional non-minimal state parameterizations for dynamical systems. One such parameterization is maximal coordinates, in which each link in a multi-body system is parameterized by its full six degrees of freedom and joints between links are modeled with algebraic constraints. Such constraints can also represent closed kinematic loops or contact with the environment. This paper investigates the difference between minimal- and maximal-coordinate LQR control laws. A case study of applying LQR to a simple pendulum and simulations comparing the basins of attraction and tracking performance of minimal- and maximal-coordinate LQR controllers suggest that maximal-coordinate LQR achieves greater robustness and improved tracking performance compared to minimal-coordinate LQR when applied to nonlinear systems.
Optimal control problems with a very large time horizon can be tackled with the Receding Horizon Control (RHC) method, which consists in solving a sequence of optimal control problems with small prediction horizon. The main result of this article is the proof of the exponential convergence (with respect to the prediction horizon) of the control generated by the RHC method towards the exact solution of the problem. The result is established for a class of infinite-dimensional linear-quadratic optimal control problems with time-independent dynamics and integral cost. Such problems satisfy the turnpike property: the optimal trajectory remains most of the time very close to the solution to the associated static optimization problem. Specific terminal cost functions, derived from the Lagrange multiplier associated with the static optimization problem, are employed in the implementation of the RHC method.
This paper is concerned with a linear quadratic optimal control for a class of singular Volterra integral equations. Under proper convexity conditions, optimal control uniquely exists, and it could be characterized via Frechet derivative of the quadratic functional in a Hilbert space or via maximum principle type necessary conditions. However, these (equivalent) characterizations have a shortcoming that the current value of the optimal control depends on the future values of the optimal state. Practically, this is not feasible. The main purpose of this paper is to obtain a causal state feedback representation of the optimal control.
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