# Stability of Riccati's Equation with Random Stationary Coefficients

## Abstract

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution (P-bar{sub t}) of this Riccati equation. This solution is attracting, in the sense that if P{sub t} is another solution, then P{sub t}-P-bar{sub t} converges to 0 exponentially fast as t tends to +{infinity} , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices.

- Authors:

- Laboratoire de Probabilites, Departement de Mathematiques Faculte des Sciences de Sfax, CP 3038 Sfax (Tunisia)

- Publication Date:

- OSTI Identifier:
- 21064283

- Resource Type:
- Journal Article

- Journal Name:
- Applied Mathematics and Optimization

- Additional Journal Information:
- Journal Volume: 40; Journal Issue: 2; Other Information: DOI: 10.1007/s002459900119; Copyright (c) 1999 Springer-Verlag New York Inc.; Article Copyright (c) Inc. 1999 Springer-Verlag New York; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0095-4616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CONTROL THEORY; HAMILTONIANS; LYAPUNOV METHOD; MATRICES; RANDOMNESS; RICCATI EQUATION; STABILITY; STOCHASTIC PROCESSES

### Citation Formats

```
Fakhfakh, S.
```*Stability of Riccati's Equation with Random Stationary Coefficients*. United States: N. p., 1999.
Web. doi:10.1007/S002459900119.

```
Fakhfakh, S.
```*Stability of Riccati's Equation with Random Stationary Coefficients*. United States. doi:10.1007/S002459900119.

```
Fakhfakh, S. Wed .
"Stability of Riccati's Equation with Random Stationary Coefficients". United States. doi:10.1007/S002459900119.
```

```
@article{osti_21064283,
```

title = {Stability of Riccati's Equation with Random Stationary Coefficients},

author = {Fakhfakh, S.},

abstractNote = {The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution (P-bar{sub t}) of this Riccati equation. This solution is attracting, in the sense that if P{sub t} is another solution, then P{sub t}-P-bar{sub t} converges to 0 exponentially fast as t tends to +{infinity} , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices.},

doi = {10.1007/S002459900119},

journal = {Applied Mathematics and Optimization},

issn = {0095-4616},

number = 2,

volume = 40,

place = {United States},

year = {1999},

month = {9}

}

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