Bellman equation

About: Bellman equation is a research topic. Over the lifetime, 5884 publications have been published within this topic receiving 135589 citations.

Sensitivity analysis in convex quadratic optimization: Simultaneous perturbation of the objective and right-hand-side vectors

Abstract: In this paper we study the behavior of Convex Quadratic Optimization problems when variation occurs simultaneously in the right-hand side vector of the constraints and in the coefficient vector of the linear term in the objective function. It is proven that the optimal value function is piecewise-quadratic. The concepts of transition point and invariancy interval are generalized to the case of simultaneous perturbation. Criteria for convexity, concavity or linearity of the optimal value function on invariancy intervals are derived. Furthermore, differentiability of the optimal value function is studied, and linear optimization problems are given to calculate the left and right derivatives. An algorithm, that is capable to compute the transition points and optimal partitions on all invariancy intervals, is outlined. We specialize the method to Linear Optimization problems and provide a practical example of simultaneous perturbation parametric quadratic optimization problem from electrical engineering.

Integro-differential equations associated with optimal stopping time of a piecewise-deterministic process

Abstract: This paper concerns the optimal stopping time problem for a piecewise deterministic process. The process has deterministic dynamics between random jumps. The as¬sociated dynamic programming equation is a variational inequality with integral and (first order) differential terms. Our main results are W4,00-existence results and probabilistic representations for the solutions of the optimal stopping time problem in bounded domains and in R. We also generalize these results to the case when the state space is “countable folds” of Euclidean space

Erratum: A Corrected Proof of the Stochastic Verification Theorem within the Framework of Viscosity Solutions

Abstract: We correct the proof of Theorem 4.1 in Gozzi, Swiech, and Zhou [SIAM J. Control Optim., 43 (2005), pp. 2009-2019] by imposing additional conditions on the viscosity subsolution $U$.

Nonlinear Optimal Control Theory

Abstract: Examples of Control Problems Introduction A Problem of Production Planning Chemical Engineering Flight Mechanics Electrical Engineering The Brachistochrone Problem An Optimal Harvesting Problem Vibration of a Nonlinear Beam Formulation of Control Problems Introduction Formulation of Problems Governed by Ordinary Differential Equations Mathematical Formulation Equivalent Formulations Isoperimetric Problems and Parameter Optimization Relationship with the Calculus of Variations Hereditary Problems Relaxed Controls Introduction The Relaxed Problem Compact Constraints Weak Compactness of Relaxed Controls Filippov's Lemma The Relaxed Problem Non-Compact Constraints The Chattering Lemma Approximation to Relaxed Controls Existence Theorems Compact Constraints Introduction Non-Existence and Non-Uniqueness of Optimal Controls Existence of Relaxed Optimal Controls Existence of Ordinary Optimal Controls Classes of Ordinary Problems Having Solutions Inertial Controllers Systems Linear in the State Variable Existence Theorems Non Compact Constraints Introduction Properties of Set Valued Maps Facts from Analysis Existence via the Cesari Property Existence without the Cesari Property Compact Constraints Revisited The Maximum Principle and Some of its Applications Introduction A Dynamic Programming Derivation of the Maximum Principle Statement of Maximum Principle An Example Relationship with the Calculus of Variations Systems Linear in the State Variable Linear Systems The Linear Time Optimal Problem Linear Plant-Quadratic Criterion Problem Proof of the Maximum Principle Introduction Penalty Proof of Necessary Conditions in Finite Dimensions The Norm of a Relaxed Control Compact Constraints Necessary Conditions for an Unconstrained Problem The epsilon-Problem The epsilon-Maximum Principle The Maximum Principle Compact Constraints Proof of Theorem 6.3.9 Proof of Theorem 6.3.12 Proof of Theorem 6.3.17 and Corollary 6.3.19 Proof of Theorem 6.3.22 Examples Introduction The Rocket Car A Non-Linear Quadratic Example A Linear Problem with Non-Convex Constraints A Relaxed Problem The Brachistochrone Problem Flight Mechanics An Optimal Harvesting Problem Rotating Antenna Example Systems Governed by Integrodifferential Systems Introduction Problem Statement Systems Linear in the State Variable Linear Systems/The Bang-Bang Principle Systems Governed by Integrodifferential Systems Linear Plant Quadratic Cost Criterion A Minimum Principle Hereditary Systems Introduction Problem Statement and Assumptions Minimum Principle Some Linear Systems Linear Plant-Quadratic Cost Infinite Dimensional Setting Bounded State Problems Introduction Statement of the Problem epsilon-Optimality Conditions Limiting Operations The Bounded State Problem for Integrodifferential Systems The Bounded State Problem for Ordinary Differential Systems Further Discussion of the Bounded State Problem Sufficiency Conditions Nonlinear Beam Problem Hamilton-Jacobi Theory Introduction Problem Formulation and Assumptions Continuity of the Value Function The Lower Dini Derivate Necessary Condition The Value as Viscosity Solution Uniqueness The Value Function as Verification Function Optimal Synthesis The Maximum Principle Bibliography Index