# Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions

## Abstract

We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples.

- Authors:

- Dipartimento di Matematica P. e A., Universita di Padova, via Belzoni 7, 35131 Padova (Italy), E-mail: acesar@math.unipd.it

- Publication Date:

- OSTI Identifier:
- 21064215

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Applied Mathematics and Optimization; Journal Volume: 53; Journal Issue: 1; Other Information: DOI: 10.1007/s00245-005-0834-1; Copyright (c) 2006 Springer; www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CONTROL THEORY; DIFFUSION; FUNCTIONS; HAMILTON-JACOBI EQUATIONS; LYAPUNOV METHOD; PROBABILITY; STABILITY; STOCHASTIC PROCESSES; VISCOSITY

### Citation Formats

```
Cesaroni, Annalisa.
```*Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions*. United States: N. p., 2006.
Web. doi:10.1007/S00245-005-0834-1.

```
Cesaroni, Annalisa.
```*Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions*. United States. doi:10.1007/S00245-005-0834-1.

```
Cesaroni, Annalisa. Sun .
"Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions". United States.
doi:10.1007/S00245-005-0834-1.
```

```
@article{osti_21064215,
```

title = {Lyapunov Stabilizability of Controlled Diffusions via a Superoptimality Principle for Viscosity Solutions},

author = {Cesaroni, Annalisa},

abstractNote = {We prove optimality principles for semicontinuous bounded viscosity solutions of Hamilton-Jacobi-Bellman equations. In particular, we provide a representation formula for viscosity supersolutions as value functions of suitable obstacle control problems. This result is applied to extend the Lyapunov direct method for stability to controlled Ito stochastic differential equations. We define the appropriate concept of the Lyapunov function to study stochastic open loop stabilizability in probability and local and global asymptotic stabilizability (or asymptotic controllability). Finally, we illustrate the theory with some examples.},

doi = {10.1007/S00245-005-0834-1},

journal = {Applied Mathematics and Optimization},

number = 1,

volume = 53,

place = {United States},

year = {Sun Jan 15 00:00:00 EST 2006},

month = {Sun Jan 15 00:00:00 EST 2006}

}

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