## Optimal and Nonlinear Control Using Multiscale Methods

The idea of wavelets can be traced back to Calderon, and Coifman and Weiss. The theory was put into a rigorous framework by a number of mathematicians in the late 1980s, such as Battle, Daubechies, Mallat and others. Upto this date the use of wavelets has been focused, to a large extend, on image and signal processing applications. More recently, the advantages of wavelets for solving partial differential equations (pde's) and integral equations have been recognized. The most significant property of wavelets, compared to other basis functions, is their ability to capture the local behavior of signals both in frequency and time. Therefore, wavelet expansions can capture very accurately a signal in both domains with very few terms (i.e., wavelet coefficients). Similarly, they are inherently adaptive in the sense that one can add or remove wavelet coefficients depending on the accuracy required, without affecting the remaining coefficients. In this work we exploit the benefits of wavelets for solving optimal control problems and for on-line denoising of signals in a feedback loop.

**Sponsors**

This project is sponsored by NSF.

**Selected Publications**

- Park, C. and Tsiotras, P., "Suboptimal Feedback Control Using a Successive Wavelet-Galerkin Algorithm'',
*Proceedings, American Control Conference,*Denver, Colorado, June 4-6, 2003, pp. 1926--1931. - Park, C. and Tsiotras, P.,"Approximations to Optimal Feedback Control Using a Successive Wavelet-Collocation Algorithm'',
*Proceedings, American Control Conference*, Denver, Colorado, June 4-6, 2003, pp. 1950--1955. - Chaplais, F., Tsiotras, P. and Jung, D., "On-Line Wavelet Denoising with Application to the Control of a Reaction Wheel
System,''
*AIAA Guidance, Navigation, and Control Conference*, Providence, RI, August 16--19, 2004. - Jain, S. and Tsiotras, P., "The Method Of Reflection For Solving The Time-Optimal Hamilton-Jacobi-Bellman Equation
On The Interval,''
*13th IEEE Mediterranean Conference on Control and Automation*, Limassol, Cyprus, June 27-29, 2005, pp. 1062-1067. - Jain, S. and Tsiotras, P.,"Solution of the Time-Optimal Hamilton-Jacobi-Bellman Equation using
Wavelets,''
*43rd IEEE Conference on Decision and Control*, Paradise Island, Bahamas, Dec. 14-17, 2004, pp. 2728-2733. - Chaplais, F., Tsiotras, P. and Jung, D., "Redundant Wavelet Filter Banks on the Half-Axis with Applications to Signal Denoising with
Small Delays,''
*43rd IEEE Conference on Decision and Control*, Paradise Island, Bahamas, Dec. 14-17, 2004, pp. 3102-3108. - Jain, S. and Tsiotras, P., "Multiresolution-Based Direct Trajectory Optimization,''
*AIAA Journal of Guidance Control, and Dynamics,*Vol. 31, No. 5, pp. 1424-1436, 2008. - Jain, S., Tsiotras, P., and Zhou, H.-M., "A
Hierarchical Multiresolution Adaptive Mesh Refinement for the Solution of Evolution PDEs,''
*SIAM Journal of Scientific Computing*, Vol. 31, No. 2, pp. 1221-1248, 2009.