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Design of Optimal Controls in Acausal LQG Problems

Arzu Ahmadova, Agamirza E. Bashirov

TL;DR

The paper addresses the design of optimal controls for acausal LQG problems, where noise processes can be delayed or wide-band, challenging the standard causal separation principle.It develops an extended separation framework that yields a two-term control structure separating a causal term based on filtered state estimates and a second term capturing acausal effects via auxiliary processes like $\alpha_t$ and backward Riccati dynamics.A series of case studies—state noise only, state and observation noise, and state with delayed white noise—demonstrates how BN and delay phenomena are incorporated via relaxing functions, infinite-dimensional Kalman filtering, and coupled Riccati equations to produce implementable optimal controls.The results provide a complete, analytically tractable design methodology for acausal LQG problems, with invariance properties to the choice of relaxing functions and clear pathways toward nonlinear extensions and practical numerical methods.

Abstract

In control theory, a system which has output depending only on the present and past values of the input is said to be causal (or nonanticipative). Respectively, a system is acausal (or non-causal) if its output depends on future inputs as well. Overall majority of literature in stochastic control theory discusses causal systems. Only a few sources indirectly concern acausal systems. In this paper, we systemize these results under main idea of acausality and present a background for designing optimal controls in acausal LQG problems.

Design of Optimal Controls in Acausal LQG Problems

TL;DR

The paper addresses the design of optimal controls for acausal LQG problems, where noise processes can be delayed or wide-band, challenging the standard causal separation principle.It develops an extended separation framework that yields a two-term control structure separating a causal term based on filtered state estimates and a second term capturing acausal effects via auxiliary processes like $\alpha_t$ and backward Riccati dynamics.A series of case studies—state noise only, state and observation noise, and state with delayed white noise—demonstrates how BN and delay phenomena are incorporated via relaxing functions, infinite-dimensional Kalman filtering, and coupled Riccati equations to produce implementable optimal controls.The results provide a complete, analytically tractable design methodology for acausal LQG problems, with invariance properties to the choice of relaxing functions and clear pathways toward nonlinear extensions and practical numerical methods.

Abstract

In control theory, a system which has output depending only on the present and past values of the input is said to be causal (or nonanticipative). Respectively, a system is acausal (or non-causal) if its output depends on future inputs as well. Overall majority of literature in stochastic control theory discusses causal systems. Only a few sources indirectly concern acausal systems. In this paper, we systemize these results under main idea of acausality and present a background for designing optimal controls in acausal LQG problems.
Paper Structure (21 sections, 3 theorems, 96 equations)

This paper contains 21 sections, 3 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. General acausality in LQG problems
  3. Admissible controls
  4. Extended separation principle
  5. Concluding remarks
  6. Case 1: State is corrupted by BN
  7. Setting of problem
  8. Optimal control
  9. Sketch of the proof
  10. Discussion
  11. Case 2: State and observations are corrupted by BN
  12. Setting of problem
  13. Optimal control
  14. Sketch of the proof
  15. Discussion
  16. ...and 6 more sections

Key Result

Lemma 2.1

Assume that $A,H,F\in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m}$, $C\in \mathbb{R}^{k\times n}$, $G\in \mathbb{R}^{m\times m}$, $H\ge 0$, $F\ge 0$, $G>0$, $w$ and $v$ are $n$- and $k$-dimensional martingales, $\varphi ^1\in L_2(0,T;L_2(\Omega ,\mathbb{R}^n))$, $\varphi ^2\in L_2(0,T;L_2( where and $x^*$ is the state process corresponding to the control $u^*$.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.1: Extended Separation Principle
  • proof