# Linear Quadratic Optimal Control Problems with Fixed Terminal States and Integral Quadratic Constraints

## TL;DR

This paper develops a method to solve linear quadratic optimal control problems with fixed terminal states and integral quadratic constraints, introducing a Riccati equation with infinite terminal value and deriving explicit optimal controls.

## Contribution

It introduces a novel Riccati equation with infinite terminal value and explicitly derives optimal controls considering both current state and target, with solution approximation techniques.

## Key findings

- Unique solvability of the Riccati equation with infinite terminal value
- Explicit optimal control formulas involving state and target feedback
- Illustrative examples demonstrating the theoretical results

## Abstract

This paper is concerned with a linear quadratic (LQ, for short) optimal control problem with fixed terminal states and integral quadratic constraints. A Riccati equation with infinite terminal value is introduced, which is uniquely solvable and whose solution can be approximated by the solution for a suitable unconstrained LQ problem with penalized terminal state. Using results from duality theory, the optimal control is explicitly derived by solving the Riccati equation together with an optimal parameter selection problem. It turns out that the optimal control is not only a feedback of the current state, but also a feedback of the target (terminal state). Some examples are presented to illustrate the theory developed.