My research interests lie at the intersection of differential equations, control theory, and numerical methods for PDEs. My most recent work is concerned with numerical methods for Hamilton-Jacobi PDEs arising in optimal control theory and the computation of center manifolds generated by periodic trajectories. I am also interested in differential geometric methods in control theory primarily in the problem of geometric controllability for control-affine systems.
These are nonlinear PDEs arising in a broad range of integral minimization problems in optimal control, computer vision, geometric optics, and classical mechanics [1,2]. The notion of viscosity solutions has been developed to settle the problem of existence and uniqueness of solutions to these nonlinear PDEs. In control theory, HJ PDEs arise in the problem of controlling an ODE
so as to minimize an integral cost
over all measurable control functions u(s). The value function
for the minimization problem satisfies a HJ PDE for the Hamiltonian function
In the literature, this HJ PDE is usually referred to as a Hamilton-Jacobi-Bellman (HJB) PDE due to its relation with the method of dynamic programming pioneered by Richard Bellman. Using a level set method, a Cauchy-Kowalevski technique, and the idea of patchy vector fields, I and co-authors [3,4,5] have been developing a numerical method to compute solutions to the HJB PDE. Below is a sample of the method via the animation of the propagation of the level sets of the value function for the classical cart-pendulum system and a 3D cubic polynomial system with two control inputs.
2D pendulum (.avi) | 3D polynomial system (.avi) |
Both animations show the nonlinearity of HJ PDEs: self-intersection of the level sets and the formation of cusps on the level sets. In both cases, the value function looses differentiability but remains Lipschitz continuous. My current work is focused on developing robust numerical algorithms to deal with these nonlinear effects and the presence of fast and slow dynamics. Another issue of interest involves the construction and/or incorporation of reliable adaptive algorithms to describe the local geometry of the level sets such as nearest neighbors information.
[1] | J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999. |
[2] | L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, 1998. |
[3] | C. Navasca, A.J. Krener, Patchy Solution of the HJB PDE, In A. Chiuso, A. Ferrante and S. Pinzoni, eds, Modeling, Estimation, and Control, Lecture Notes in Control and Information Sciences, 364, 251-270, 2007. |
[4] | C.O. Aguilar, A.J. Krener, High-order numerical solutions to Bellman's equation of optimal control, Proc. American Control Conference, 2012, pp. 1832--1837. |
[5] | C.O. Aguilar, A.J. Krener, Numerical solutions to the Bellman equation of optimal control, Journal of Optimization Theory and Applications, to appear, 2013. |
Invariant manifolds provide crucial insight into the overall qualitative behavior of a dynamical system. For example, an important feature of invariant manifolds is the simplification resulting from reduction of dimension as they organize the dynamics into smaller regions of the configuration space. Recently, there has been an active effort in developing numerical methods to compute (un)stable invariant manifolds of vector fields using geodesic level sets, BVP continuation of trajectories, computation of ``fat'' trajectories, and PDE formulations [6]. The computation of center manifolds has received far less attention partially due to the subtle properties regarding existence and regularity of center manifolds [7]. In control theory, the need to compute a center manifold arises in the problem of stabilizing a reference trajectory of the configuration variables. Through center manifold reduction, the trajectory stabilization problem can be solved by computing the solution φ to the invariant manifold PDE
for the coupled ODEs
The w-dynamics act like a forcing term in the z-dynamics and generate the desired reference trajectory to be stabilized. When the w-dynamics are 2-dimensional and neutrally stable, and the z-dynamics are hyperbolic, the invariant (center) manifold is generated by a real analytic one-parameter family of periodic trajectories [8]. In [9], I and A.J. Krener developed a BVP continuation algorithm to compute the center manifold determined by the graph of φ. Our work naturally lead to an investigation of extending our algorithm to higher-dimensional forcing terms [10]. My current work in this area is to extend the BVP continuation algorithm in [9] to isochronous forcing terms that possess a first integral and a natural choice of these systems are Hamiltonian isochronous systems.
[6] | B. Krauskopf, H.M. Osinga, E.J. Deodel, M.E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Delintz, O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Int. Journal of Bifurcation and Chaos, 15, 763-791, 2005. |
[7] | J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (2), 431-469, 1985. |
[8] | B. Aulbach, A classical approach to the analyticity problem of center manifolds, Journal of Applied Mathematics and Physics, 36(1), 1-23, 1985. |
[9] | C.O. Aguilar, A.J. Krener, Patchy solution of a Francis-Byrnes-Isidori partial differential equation,Int. Journal of Robust and Nonlinear Control, Vol. 23, No. 9, pp. 1046-1061, 2013. |
[10] | C.O. Aguilar, On the existence and uniqueness of solutions to the output regulator equations for periodic exosystems, Systems and Control Letters, 61, 702-706, 2012. |
Given the controlled differential equation
and a positive time T, the reachable set R(T) consists of all the final states x(T) obtained by following the trajectories of the above control system as one varies the control function u(t) over some prescribed set of functions. The vector field f(x,u) determines the allowable infinitesimal motions from the point x and usually these are a strict subset of the full tangent space at x. The classic example in control theory is parallel parking of a vehicle. At an initial configuration x assumed to be parallel with the curb, one can move only in a direction parallel with the curb and along directions tangent to a circle determined by the turning radius. At such an initial configuration, one cannot move in a direction perpendicular to the curb. However, as is common experience, by switching between these two allowable directions, it is possible to move the vehicle in such a way that its final position is a perpendicular shift of its initial condition. My work in the area of controllability centers on the notion of small-time local controllability (STLC) [11,12,13]. Recently [13], I considered the STLC property for a class of polynomial control-affine systems and determined a feedback invariant sufficient condition for STLC using Butcher rooted trees. It would be interesting to extend this work and develop a complete feedback invariant STLC characterization for these polynomial systems.
[11] | C.O. Aguilar, Local controllability of affine distributions Ph.D. thesis, Queen's University, defended Dec 2009. |
[12] | C.O. Aguilar and A.D. Lewis, Small-time local controllability for a class of homogeneous systems, SIAM Journal on Control and Optimization, Vol. 50, No. 3, pp. 1502-1517, 2012. (journal version) |
[13] | C.O. Aguilar, Local controllability of control-affine systems with quadratic drift and constant control-input vector fields, Proc. 51th IEEE Conf. Decision and Control, pp. 1877--1882, 2012. |