We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by applications in machine learning through deep residual neural networks, which may be fit within our setting. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.
Turnpike phenomena of nonlinear port-Hamiltonian descriptor systems under minimal energy supply are studied. Under assumptions on the smoothness of the system nonlinearities, it is shown that the optimal control problem is dissipative with respect to a manifold. Then, under controllability assumptions, it is shown that the optimal control problem exhibits a manifold turnpike property.
This paper presents, using dynamical system theory, a framework for investigating the turnpike property in nonlinear optimal control. First, it is shown that a turnpike-like property appears in general dynamical systems with hyperbolic equilibrium and then, apply it to optimal control problems to obtain sufficient conditions for the turnpike occurs. The approach taken is geometric and gives insights for the behaviors of controlled trajectories, allowing us to find simpler proofs for existing results on the turnpike properties. Attempts to remove smallness restrictions for initial and target st…
We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very ef…
In this work we derive an interval turnpike result for adjoints of finite- and infinite-dimensional nonlinear optimal control problems under the assumption of an interval turnpike on states and controls. We consider stabilizable dynamics governed by a generator of a semigroup with finite-dimensional unstable part satisfying a spectral decomposition condition and show the desired turnpike property under continuity assumptions on the first-order optimality conditions. We further give stronger estimates for analytic semigroups and provide a numerical example with a boundary controlled semilinear …
Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems
Université Paris-Saclay, CNRS, CentraleSupélec（巴黎萨克雷大学）；Università degli Studi di Padova（帕多瓦大学）；Sorbonne Université, Université Paris Cité, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎西岱大学）
Turnpike properties have been established long time ago in finite-dimensional optimal control problems arising in econometry. They refer to the fact that, under quite general assumptions, the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces, the first and the last of which being transient short-time arcs, and the middle piece being a long-time arc staying exponentially close to the optimal steady-state solution of an associated static optimal control problem. We provide in this paper a general version of a turnpike theorem, valuab…
The paper addresses an existence problem for infinite horizon optimal control when the system under control is exponentially stabilizable or stable. Classes of nonlinear control systems for which infinite horizon optimal controls exist are identified in terms of stability, stabilizability, detectability and growth conditions. The result then applies to estimate the existence region of stable manifolds in the associated Hamiltonian systems. Applications of the results also include the analysis for turnpike property in nonlinear finite horizon optimal control problems by a geometric approach.
Global $H_\infty$ Consensus of Multi-Agent Systems with Lipschitz Nonlinear Dynamics
School of Automation, Beijing Institute of Technology（北京理工大学自动化学院）；School of Electrical and Electronic Engineering, Nanyang Technological University（南洋理工大学电气与电子工程学院）
Numerous state-feedback and observer designs for nonlinear dynamic systems (NDS) have been developed in the past three decades. These designs assume that NDS nonlinearities satisfy one of the following function set classifications: bounded Jacobian, Lipschitz continuity, one-sided Lipschitz, quadratic inner-boundedness, and quadratic boundedness. These function sets are characterized by constant scalars or matrices bounding the NDS' nonlinearities. These constants (i) depend on the NDS' operating region, topology, and parameters, and (ii) are utilized to synthesize observer/controller gains. U…
Consensus of Multi-Agent Systems with General Linear and Lipschitz Nonlinear Dynamics Using Distributed Adaptive Protocols
School of Automation, Beijing Institute of Technology（北京理工大学自动化学院）；Department of Electrical and Computer Engineering, Utah State University（犹他州立大学电气与计算机工程系）
We study state estimation for nonlinear differential-algebraic systems, where the nonlinearity satisfies a Lipschitz condition or a generalized monotonicity condition or a combination of these. The presented observer design unifies earlier approaches and extends the standard Luenberger type observer design. The design parameters of the observer can be obtained from the solution of a linear matrix inequality restricted to a subspace determined by the Wong sequences. Some illustrative examples and a comparative discussion are given.
Nonlinear dynamic systems can be classified into various classes depending on the modeled nonlinearity. These classes include Lipschitz, bounded Jacobian, one-sided Lipschitz (OSL), and quadratically inner-bounded (QIB). Such classes essentially yield bounding constants characterizing the nonlinearity. This is then used to design observers and controllers through Riccati equations or matrix inequalities. While analytical expressions for bounding constants of Lipschitz and bounded Jacobian nonlinearity are studied in the literature, OSL and QIB classes are not thoroughly analyzed---computationa…
Drinking water distribution networks (WDN) are large-scale, dynamic systems spanning large geographic areas. Water networks include various components such as junctions, reservoirs, tanks, pipes, pumps, and valves. Hydraulic models for these components depicting mass and energy balance form nonlinear algebraic differential equations (NDAE). While control theoretic studies have been thoroughly explored for other complex infrastructure such as power and transportation systems, little is understood or even investigated for feedback control and state estimation problems for the NDAE models of WDN.…
This paper presents an inverse problem for the nonlinear 1-d Kuramoto-Sivashinsky (K-S) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and at some positive time everywhere. Uniqueness and Lipschitz stability for this inverse problem are proven with the Bukhgeim-Klibanov method. The proof is based on a global Carleman estimate for the linearized K-S equation.
We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz informa…
When the objective function is not locally Lipschitz, constraint qualifications are no longer sufficient for Karush-Kuhn-Tucker (KKT) conditions to hold at a local minimizer, let alone ensuring an exact penalization. In this paper, we extend quasi-normality and relaxed constant positive linear dependence (RCPLD) condition to allow the non-Lipschitzness of the objective function and show that they are sufficient for KKT conditions to be necessary for optimality. Moreover, we derive exact penalization results for the following two special cases. When the non-Lipschitz term in the objective funct…
Although an appropriate choice of measured state variables may ensure observability, designing state observers for the state estimation of epidemic models remains a challenging task. Epidemic spread is a nonlinear process, often modeled as the law of mass action, which is of a quadratic form; thus, on a compact domain, its Lipschitz constant turns out to be local and relatively large, which renders the Lipschitz-based design criteria of existing observer architectures infeasible. In this paper, a novel observer architecture is proposed for the state estimation of a class of nonlinear systems t…
In this paper, we consider the problem of automatically designing a Rectified Linear Unit (ReLU) Neural Network (NN) architecture (number of layers and number of neurons per layer) with the assurance that it is sufficiently parametrized to control a nonlinear system; i.e. control the system to satisfy a given formal specification. This is unlike current techniques, which provide no assurances on the resultant architecture. Moreover, our approach requires only limited knowledge of the underlying nonlinear system and specification. We assume only that the specification can be satisfied by a Lips…
Asymptotic stationarity and regularity conditions turned out to be quite useful to study the qualitative properties of numerical solution methods for standard nonlinear and complementarity-constrained programs. In this paper, we first extend these notions to nonlinear optimization problems with nonsmooth but Lipschitzian data functions in order to find reasonable notions of asymptotic stationarity and regularity in terms of Clarke's and Mordukhovich's subdifferential construction. Particularly, we compare the associated novel asymptotic constraint qualifications with already existing ones. The…
In this paper we prove two results regarding reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem"). First we show that phase retrievability as an algebraic property implies that nonlinear maps are bi-Lipschitz with respect to appropriate metrics on the quotient space. Second we prove that reconstruction can be performed using Lipschitz continuous maps. Specifically we show that when nonlinear analysis maps $α,β:\hat{H}\rightarrow R^m$ are injective, with $α(x)=(|\langle x,f_k\rangle |)_{k=1}^m$ and $β(x)=(|\langle x,f_k \rangle|^2)_{k=1}^m$, where $\{f_…
We develop a theoretical foundation for the application of Nesterov's accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requ…
This paper studies stability aspects of solutions of parametric mathematical programs and generalized equations, respectively, with disjunctive constraints. We present sufficient conditions that, under some constraint qualifications ensuring metric subregularity of the constraint mapping, continuity results of upper Lipschitz and upper Hölder type, respectively, hold. Furthermore, we apply the above results to parametric mathematical programs with equilibrium constraints and demonstrate, how some classical results for the nonlinear programming problem can be recovered and even improved by our …
We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a const…
