| The
quasiconvex envelope through first-order partial
differential equations which characterize
quasiconvexity of
nonsmooth functions ABSTRACT: Necessary and sufficient conditions for quasiconvexity, also called level-set convexity, of a function are given in terms of first-order partial differential equations. Solutions to the equations are understood in the viscosity sense and the conditions apply to nonsmooth and semicontinuous functions. A comparison principle, implying uniqueness of solutions, is shown for a related partial differential equation. This equation is then used in an iterative construction of the quasiconvex envelope of a function. The results are then extended to robustly quasiconvex functions, that is, functions which are quasiconvex under small linear perturbations. |
| Quasiconvex
functions and nonlinear PDEs ABSTRACT: A second order characterization of functions which have convex level sets (quasiconvex functions) results in the operator $L_0(Du,D^2u)=\min\{v\.D^2u\,v^T\;|\;|v|=1,|v\.Du|=0\}.$ In two dimensions this is the mean curvature operator and in any dimension $L_0(Du,D^2u)/|Du|$ is the first principal curvature of the surface $S=u^{-1}(c).$ Our main results include a comparison principle for $L_0(Du,D^2u)=g$ when $g \geq C_g>0$ and a comparison principle for \textit{quasiconvex} solutions of $L_0(Du,D^2u)=0.$ A more regular version of $L_0$ is introduced, namely $L_\a(Du,D^2u)=\min\{v\.D^2u\,v^T\;|\;|v|=1,|v\.Du| \leq \a\}$ which characterizes functions which remain quasiconvex under small linear perturbations. A comparison principle is proved for $L_\a$. A representation result using stochastic control is also given and we consider the obstacle problems for $L_0$ and $L_\a$. |
| The optimal value
and optimal solutions of the proximal average of
convex functions ABSTRACT: The proximal average of a finite collection of convex functions is a parameterized convex function that provides a continuous transformation between the convex functions in the collection. This paper analyzes the dependence of the optimal value and the minimizers of the proximal average on the weighting parameter. Concavity of the optimal value is established and implies further regularity properties of the optimal value. Boundedness, outer semicontinuity, single-valuedness, continuity, and Lipschitz continuity of the minimizer mapping are concluded under various assumptions. Sharp minimizers are given further attention. Several examples are given to illustrate our results. |
| Set-valued Lyapunov
functions for difference inclusions ABSTRACT: The paper relates set-valued Lyapunov functions to pointwise asymptotic stability in systems described by a difference inclusion. Pointwise asymptotic stability of a set is a property which requires that each point of the set be Lyapunov stable and that every solution to the inclusion, from a neighborhood of the set, be convergent and have the limit in the set. Weak set-valued Lyapunov functions are shown, via an argument resembling an invariance principle, to imply this property. Strict set-valued Lyapunov functions are shown, in the spirit of converse Lyapunov results, to always exist for closed sets that are pointwise asymptotically stable. |
| The proximal average for
saddle functions and its symmetry properties with
respect to partial and saddle conjugacy
ABSTRACT: The concept of the proximal average for convex functions is extended to saddle functions. Self-duality of the proximal average is established with respect to partial conjugacy, which pairs a convex function with a saddle function, and saddle function conjugacy, which pairs a saddle function with a saddle function. |
| Best response
dynamics for continuous games with E.N.
Barron and R.R. Jensen ABSTRACT: We extend the convergence result of Hofbauer and Sorin \cite{Hof} for the best response differential inclusions coming from a nonconcave, nonconvex continuous payoff function $U(x,y)$. A counterexample shows that convergence to a Nash equilibrium may not be true if we attempt to generalize the result to a three person nonzero sum game. |
| Pre-asymptotic
stability and homogeneous approximations of hybrid
dynamical systems , with A.R. Teel. ABSTRACT: Hybrid dynamical systems are systems that combine features of continuous-time dynamical systems and discrete-time dynamical systems, and can be modeled by a combination of differential equations or inclusions, difference equations or inclusions, and constraints. Pre-asymptotic stability is a concept that results from separating the conditions that asymptotic stability places on the behavior of solutions from issues related to existence of solutions. In this paper, techniques for approximating hybrid dynamical systems that generalize classical linearization techniques are proposed. The approximation techniques involve linearization, tangent cones, homogeneous approximations of functions and set-valued mappings, and tangent homogeneous cones, where homogeneity is considered with respect to general dilations. The main results deduce pre-asymptotic stability of an equilibrium point for a hybrid dynamical system from pre-asymptotic stability of the equilibrium point for an approximate system. Further results relate the degree of homogeneity of a hybrid system to the Zeno phenomenon that can appear in the solutions of the system. |
| Smooth patchy
control Lyapunov functions , with C. Prieur and A.R.
Teel. ABSTRACT: A smooth patchy control Lyapunov function for a nonlinear system consists of an ordered family of smooth local control Lyapunov functions, whose open domains form a locally finite cover of the state space of the system, and which satisfy certain further increase or decrease conditions. We prove that such a control Lyapunov function exists for any asymptotically controllable nonlinear system. We also show a construction, based on such a control Lyapunov function, of a stabilizing hybrid feedback that is robust to measurement noise. |
| Direct design of robustly
asymptotically stabilizing hybrid feedback, with A. Teel. ABSTRACT: A direct construction of a stabilizing hybrid feedback that is robust to general measurement error is given for a general nonlinear control system that is asymptotically controllable to a compact set. |
| Relaxation
theorems for hybrid inclusions, with C. Cai and A.R. Teel.
ABSTRACT: The Filippov-Wazewski Relaxation Theorem describes when the set of solutions to a differential inclusion is dense in the set of solutions to the relaxed (convexified) differential inclusion. This paper establishes relaxation results for a broad range of hybrid systems which combine differential inclusions, difference inclusions, and constraints on the continuous and discrete motions induced by these inclusions. The relaxation results are used to deduce continuous dependence on initial conditions of the sets of solutions to hybrid systems. |
| Invariance
principles for switching systems via hybrid systems
techniques , with R. Sanfelice and
A. Teel. ABSTRACT: Invariance principles and sufficient conditions for asymptotic stability for switching systems are given. Multiple Lyapunov-like functions are used, and dwell-time, persistent dwell-time, and weak dwell-time switching signals are considered. The invariance principles are derived from general invariance principles for hybrid systems. Asymptotic stability is concluded under observability assumptions or common bounds on the Lyapunov-like functions. |
| Generalized solutions to
hybrid dynamical systems , with R. Sanfelice and A. Teel. ABSTRACT: Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise is present in the system. |
| The proximal average:
basic theory,
with H. Bauschke, Y. Lucet, and X. Wang. ABSTRACT: The recently introduced proximal average of two convex functions is a convex function with many useful properties. In this paper, we introduce and systematically study the proximal average for finitely many convex functions. The basic properties of the proximal average with respect to the standard convex-analytical notions (domain, Fenchel conjugate, subdifferential, proximal mapping, and others) are provided and illustrated by several examples. |
| Local strong
convexity and local Lipschitz continuity of the
gradient of convex functions, with R.T. Rockafellar. ABSTRACT: Given a pair of convex conjugate functions f and f*, we investigate the relationship between local Lipschitz continuity of the gradient of f and local strong convexity properties of f*. |
| Self-dual smoothing
of convex and saddle functions . ABSTRACT: We show that any convex function can be approximated by a family of differentiable with Lipschitz continuous gradient and strongly convex approximates in a ``self-dual'' way: the conjugate of each approximate is the approximate of the conjugate of the original function. The approximation technique extends to saddle functions, and is self-dual with respect to saddle function conjugacy and also partial conjugacy that relates saddle functions to convex functions. |
| Smooth Lyapunov
functions for hybrid systems. Part II:
(Pre-)asymptotically stable compact sets , with C. Cai and A. Teel. ABSTRACT: It is shown that (pre-)asymptotic stability, which generalizes asymptotic stability, of a compact set for a hybrid system satisfying certain basic growth and closedness conditions is equivalent to the existence of a smooth Lyapunov function. This result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small, state-dependent perturbations. As a special case, we state a converse Lyapunov theorem for systems with logic variables and use this result to establish input-to-state stabilization using hybrid feedback control. The converse Lyapunov theorems are also used to establish semi-global practical robustness to slowly-varying, weakly-jumping parameters, to temporal regularization, to the insertion of jumps according to an ``average dwell-time'' rule, and to the insertion of flow according to a ``reverse average dwell-time'' rule. |
| Invariance principles
for hybrid systems with connections to detectability
and asymptotic stability , with R. Sanfelice and A. Teel. ABSTRACT: Versions of LaSalle's invariance principle for general hybrid systems are presented. One version involves the use of a nonincreasing function, like in the original work of LaSalle. The other version involves ``meagreness'' conditions as recently introduced by Ryan and co-authors. These principles characterize asymptotic convergence of bounded hybrid trajectories to weakly invariant sets. Differences between the two versions of the invariance principle are discussed and illustrated by examples. A detectability property is used to locate a set in which the Omega-limit set of a trajectory is contained. Next, it is shown how the invariance principles can be used to certify asymptotic stability in hybrid systems. Lyapunov and Krasovskii theorems for hybrid systems are included. Finally, examples are given to illustrate the results. |
| Hybrid feedback control and
robust stabilization of nonlinear systems , with C. Prieur and A. Teel. ABSTRACT: We show that a nonlinear system that is asymptotically controllable to a compact set can be stabilized by a hybrid feedback that is robust to small measurement noise, actuator error, and external disturbances. The construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework, and giving particular attention to semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system. The robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties. Auxiliary results show uniformity of convergence and of overshoots for hybrid systems, and give a $\mathcal{KL}$ characterization of asymptotic stability of compact sets. An alternate construction of a hybrid feedback from a patchy one, dealing with the robustness directly, is also given. |
| Smooth Lyapunov
functions for hybrid systems. Part I: Existence is
equivalent to robustness , with C. Cai and A. Teel. ABSTRACT: For a broad class of hybrid systems, we establish the equivalence between robustness of stability with respect to two measures and a characterization of this property in terms of a smooth Lyapunov function. This result unifies and generalizes previous results for differential and difference inclusions with outer semicontinuous and locally bounded right-hand sides. Furthermore, we give an equivalent description of forward completeness of a hybrid system in terms of a smooth Lyapunov-like function. |
| Continuous Time
constrained linear quadratic regulator -- convex
duality approach , with M. Subbotin. ABSTRACT: A continuous time infinite horizon linear quadratic regulator with input constraints is studied. On the theoretical side, optimality conditions, both in the open loop and feedback form, are shown together with smoothness of the value function and local Lipschitz continuity of the optimal feedback. Arguments are self-contained, rely on basic ideas of convex conjugacy, and in particular, construct and use a dual optimal control problem. On the practical side, a method of calculating the optimal and stabilizing feedback without relying on discrete optimization is outlined. |
| Conjugate convex
Lyapunov functions for dual linear differential
inclusions ,
with A. Teel, T. Hu, and Z. Lin. ABSTRACT: Tools from convex analysis are used to show how stability properties translate when passing from a linear differential inclusion (LDI) to its dual. In particular, it is proved that a convex, positive definite function is a Lyapunov function for an LDI if and only if its convex conjugate is a Lyapunov function for the LDI's dual. Examples show how such duality effectively doubles the number of tools available for assessing stability of LDIs. |
| Solutions to hybrid
inclusions via set and graphical convergence with
stability theory applications , with A. Teel. ABSTRACT: Motivated by questions in stability theory for hybrid dynamical systems, we establish some fundamental properties of the set of solutions to such systems. Using the notion of a hybrid time domain and general results on set and graphical convergence, we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and "upper semicontinuous" with respect to initial conditions and system perturbations. The latter means that each solution to the system under perturbations is close to some solution of the unperturbed system on a compact hybrid time domain. The general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets. These results parallel what is already known for differential inclusions and difference inclusions. For example, the basin of attraction for a compact attractor is (relatively) open, the attractivity is uniform from compact subsets of the basin of attraction, and asymptotic stability is robust with respect to small perturbations. |
| Conjugate Lyapunov
functions for saturated linear systems , with T. Hu, A. Teel, Z.
Lin. ABSTRACT: We use a recently developed duality theory for linear differential inclusions (LDIs) to enhance the stability analysis of systems with saturation nonlinearities. Based on the duality theory, the condition of stability for a LDI in terms of one Lyapunov function can be easily derived from that in terms of a Lyapunov function conjugate to the original one in the sense of convex analysis. This paper uses a particular conjugate pair, the convex hull of quadratics and the maximum of quadratics, along with their dual relationship, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a LDI system with an effective approach which enables optimization on the local LDI description. Furthermore, the optimization on the LDI description can be tightly integrated with the optimization of the Lyapunov functions, thus leading to a single optimization problem with the objective of maximizing the estimation of the domain of attraction. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. The analysis methods are then extended to address the problem of designing feedback laws to enlarge the domain of attraction. A numerical example demonstrates that the estimation of the domain of attraction by this paper's methods drastically improve those by the earlier methods. |
| Stabilizing a linear
systems with saturation through optimal control. ABSTRACT: We construct a continuous feedback for a saturated system \xdot(t)=Ax(t)+B\sigma(u(t)). The feedback renders the system asymptotically stable on the whole set of states that can be driven to 0 with an open-loop control. The trajectories of the resulting closed-loop system are optimal for an auxiliary optimal control problem with a convex cost and linear dynamics. The value function for the auxiliary problem, which we show to be differentiable, serves as a Lyapunov function for the saturated system. Relating the saturated system, which is nonlinear, to an optimal control problem with linear dynamics is possible thanks to the monotone structure of saturation. |
| Duality and uniqueness of
convex solutions to stationary Hamilton-Jacobi
equations. ABSTRACT: Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations of primal and dual value functions and their conjugacy are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points. |
| Convex optimal
control problems with smooth Hamiltonians.
ABSTRACT: Fully convex optimal control problems with continuously differentiable Hamiltonians, but possibly nonsmooth and infinite-valued running costs, are studied. Smoothness of the value function and Lipschitz continuity of the optimal feedback is concluded. Convex analysis is employed to describe a wide class of extended piecewise linear-quadratic control problems, satisfying the regularity assumption on the Hamiltonian. This class encompasses several extensions of the classical linear-quadratic regulator, allowing for unbounded control sets, state-dependent constraints, and various quadratic penalties. |
| Regularity of the
optimal feedback and the value function in convex
problems of optimal control . ABSTRACT: An optimal feedback mapping, leading to necessary and sufficient conditions for optimality in terms of a closed-loop differential inclusion, is derived in the setting of fully convex generalized problems of Bolza. Results are translated to teh formal of control problems with linear dynamics and convex costs. Properties of the feedback mapping, with focus on single-valuedness and continuity, are analyzed through those of the value function and the Hamiltonian. Conditions guaranteeing differentiability of the value function are obtained through the analysis of its subdifferential as a maximal monotone operator and of the generalized Hamiltonian dynamics. |
| Notions of relative
interior in Banach spaces. ABSTRACT: Extensions to a Banach space of the equivalent notions of relatively absorbing, non-support, and relative interior points of a convex set in $\reals^n$ are presented. The relations between these extensions are studied, and their basic calculus rules are developed. Several explicit examples and counterexamples in general Banach spaces are given; and the tools for development of further examples are explained. Various implications for infinite dimensional optimization are highlighted. |
| Planar generalized
Hamiltonian systems with large saddle sets . ABSTRACT: Two-dimensional Hamiltonian systems are analyzed, for Hamiltonians concave in one variable and convex in the other, nonsmooth, and possessing several saddle points. Uniqueness of solutions of such a system is studied, along with convergence properties of the bounded solutions, and the structure of level sets of the Hamiltonian. Motivation comes from one-dimensional infinite-horizon convex problems of optimal control and their duals, in particular from questions about conjugacy of the associated value functions. |
| Generalized
conjugacy in Hamilton-Jacobi theory for fully convex
Lagrangians. ABSTRACT: Control problems with fully convex Lagrangians and convex initial costs are considered. Generalized conjugacy and envelope representation in terms of a dualizing kernel are employed to recover the initial cost from the value function at some fixed future time, leading to a generalization of the cancellation rule for inf-convolution. Such recovery is possible subject to persistence of trajectories of a generalized Hamiltonian system, associated with the Lagrangian. Global analysis of Hamiltonian trajectories is carried out, leading to conditions on the Hamiltonian, and the corresponding Lagrangian, guaranteeing persistence of the trajectories. |
| Convexity in zero-sum
differential games.
. ABSTRACT: A new approach to two-player zero-sum differential games with convex-concave cost function is presented. It employs the tools of convex and variational analysis. A necessary and sufficient condition on controls to be an open-loop saddle point of the game is given. Explicit formulas for saddle controls are derived, in terms of the subdifferential of the function conjugate to the cost. Existence of saddle controls is concluded under very general assumptions. A Hamiltonian inclusion, new to the field of differential games, is shown to describe equilibrium trajectories of the game. |
| Sufficient condition for
stability of an L^2-angle. ABSTRACT: In this paper we prove a theorem about angles between certain sequences of subspaces in a Hilbert space and apply it to give a partial answer to the question of stability of an L^2-angle, stated in [4] |