Cesar O. Aguilar - Research

My research interests lie at the intersection of differential equations, control theory, and numerical methods. In my PhD thesis, advised by Andrew D. Lewis, I studied the differential geometric controllability problem and derived sufficient and necessary conditions for small-time local controllability for a class of homogeneous control systems. In my post-doc work, advised by Arthur J. Krener, I developed numerical methods to compute solutions to Bellman's dynamic programming equation arising in the optimal stabilization problem for nonlinear discrete-time systems, and I also developed a numerical method to compute solutions to periodic center manifolds arising in the output tracking problem for continuous-time nonlinear systems. Recently, and in collaboration with Bahman Gharesifard, I have been working on the controllability problem for multi-agent networked control systems or graphs. More details of my research work is provided below.

Co-authors: Sue Ann Campbell, Bahman Gharesifard, Arthur J. Krener, Andrew D. Lewis, Kirsten Morris
Software and packages I use in my research: SciPy Python, NetworkX, nauty, Armadillo C++ , CUDA C

Controllability of Networked Dynamical Control Systems

NSF ECCS Grant No. 1509966

Networks are increasingly becoming a useful tool to model complex dynamic behavior in many branches of science and engineering. A networked dynamical system is a collection of dynamic units (e.g. ODEs) that are able to communicate and share information only locally with neighboring systems. The shared information between systems can affect the dynamics of each individual unit and the global dynamics. The communication network between units determines a graph (as in graph theory) and an inherent problem is to determine in graph-theoretic terms how the graph structure affects global dynamic behavior. From a control-theoretic standpoint, one is interested in altering the natural dynamics of the networked system to achieve a desired global task such as state transfer, synchronization, sensoring, etc. A very practical problem is to understand how to properly select only a subset of the units to act as controlling agents that will be used to accomplish the desired task. Although the property of controllability is mathematically generic, many real-world networks possess a high-level of structure such as symmetries, regularity, tree structures, etc. Such structures impose significant constraints on the selection of agents to control the overall network.

Below are .txt files containing all regular connected graphs on vertices from n = 6 to n = 14. The graphs are in nauty g6 format and in Maple format as an adjacency matrix (up to n = 12). For n = 14, the graphs were separated by regularity r = 2,3, ..., 13. The files were generated using genreg and modifying the output to g6 format.

Regular graphs from n = 6 to n = 14 vertices
# Vertices g6 format Maple format # Regular graphs
6 regular_graphs6.txt regular_graphs6_Maple.txt 5
7 regular_graphs7.txt regular_graphs7_Maple.txt 4
8 regular_graphs8.txt regular_graphs8_Maple.txt 17
9 regular_graphs9.txt regular_graphs9_Maple.txt 22
10 regular_graphs10.txt regular_graphs10_Maple.txt 167
11 regular_graphs11.txt regular_graphs11_Maple.txt 539
12 regular_graphs12.txt regular_graphs12_Maple.txt 18979
13 regular_graphs13.txt 389436
14 regular_graphs14_r2.txt 1
14 regular_graphs14_r3.txt 509
14 regular_graphs14_r4.txt 88168
14 regular_graphs14_r5.txt (59 MB) 3459383
14 regular_graphs14_r6.txt (371 MB) 21609300
14 regular_graphs14_r7.txt (371 MB) 21609301
14 regular_graphs14_r8.txt (59 MB) 3459386
14 regular_graphs14_r9.txt 88193
14 regular_graphs14_r10.txt 540
14 regular_graphs14_r11.txt 13
14 regular_graphs14_r12.txt 1
14 regular_graphs14_r13.txt 1

g6ToAdj.py - a Python function converting a graph in g6 format to an adjacency matrix as a numpy array.

For more data on regular graphs visit the genreg homepage.

This research is supported by NSF ECCS Grant No. 1509966


C.O. Aguilar, B. Gharesifard, Laplacian controllability classes for threshold graphs, Linear Algebra and Its Applications, Vol. 471, pp. 575-586, 2015. (journal version)


C.O. Aguilar, B. Gharesifard, Graph controllability classes for the Laplacian leader-follower dynamics, IEEE Transactions on Automatic Control, Vol. 60, No. 6, pp. 1611-1623, 2015. (journal version)


C.O. Aguilar, B. Gharesifard, Almost equitable partitions and new necessary conditions for network controllability, Automatica, accepted, 2016.

Hamilton-Jacobi PDEs

Hamilton-Jacobi equations 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-animation 3D-levelset-animation
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.


J.A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999.


L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, AMS, 1998.


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.


C.O. Aguilar, A.J. Krener, High-order numerical solutions to Bellman's equation of optimal control, Proceedings of the American Control Conference, 2012, pp. 1832--1837. (proceedings version)


C.O. Aguilar, A.J. Krener, Numerical solutions to the Bellman equation of optimal control, Journal of Optimization Theory and Applications, Vol. 160, No. 2, pp. 527-552, 2014. (journal version)

Periodic Center Manifolds

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 [1]. The computation of center manifolds has received far less attention partially due to the subtle properties regarding existence and regularity of center manifolds [2]. 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 [3]. In [4], 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 [5]. My current work in this area is to extend the BVP continuation algorithm in [4] to isochronous forcing terms that possess a first integral and a natural choice of these systems are Hamiltonian isochronous systems.


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.


J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289 (2), 431-469, 1985.


B. Aulbach, A classical approach to the analyticity problem of center manifolds, Journal of Applied Mathematics and Physics, 36(1), 1-23, 1985.


C.O. Aguilar, A.J. Krener, Patchy solution of a Francis-Byrnes-Isidori partial differential equation, International Journal of Robust and Nonlinear Control, Vol. 23, No. 9, pp. 1046-1061, 2013. (journal version)


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. (journal version)


C.O. Aguilar, A.J. Krener, Piecewise smooth solutions to the nonlinear output regulation PDE, Proc. American Control Conference , 2011, pp. 1426-1427. (proceedings version)

Nonlinear Controllability

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) [1,2,3]. In [3], 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.


C.O. Aguilar, Local controllability of affine distributions Ph.D. thesis, Queen's University, defended Dec 2009.


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)


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. ( proceedings version)