Delfim F. M. Torres
We prove a Noether-type symmetry theorem for invariant optimal control problems with unrestricted controls. The result establishes weak conservation laws along all the minimizers of the problems, including those minimizers which do not satisfy the Pontryagin Maximum Principle.
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Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a …
Optimal control remains as one of the most versatile frameworks in systems theory, enabling applications ranging from classical robust control to real-time safe operation of fleets of vehicles. While some optimal control problems can be efficiently solved using algebraic or convex methods, most general forms of optimal control must be solved using memory- expensive numerical methods. In this paper we present a theoretical formulation and a corresponding numerical algorithm that can find Pontryagin- optimal inputs for general dynamical systems by using a direct method. Pontryagin-optimal inputs…
We obtain a method to compute effective first integrals by combining Noether's principle with the Kozlov-Kolesnikov integrability theorem. A sufficient condition for the integrability by quadratures of optimal control problems with controls taking values on open sets is obtained. We illustrate our approach on some problems taken from the literature. An alternative proof of the integrability of the sub-Riemannian nilpotent Lie group of type (2,3,5) is also given.
We provide a new, simpler, and more direct proof of the well known fact that for autonomous optimal control problems the Pontryagin extremals evolve on a level surface of the respective Pontryagin Hamiltonian.
Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach
本文用拉格朗日方法研究确定性平均场类型自由端点最优控制问题，推导庞特里亚金极大值原理，并证明Kantorovich与Eulerian形式下的局部极小值对应，最后以线性二次调节器为例说明最优策略为线性反馈。
This paper concerns a class of infinite horizon optimal control problems with state constraints. By extending the needle variation method to the infinite horizon case we obtain a complete set of necessary optimality conditions for a strong local minimizer in form of the Pontryagin maximum principle and the validity of several transversality conditions at infinity. To this purpose, we develop an approach in the space of continuous functions converging at infinity. The considerations including new results on optimality and transversality conditions, on Banach spaces of continuous functions, on t…
Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreove…
In this paper, we analyse control affine optimal control problems with a cost functional involving the absolute value of the control. The Pontryagin extremals associated with such systems are given by (possible) concatenations of bang arcs with singular arcs and with inactivated arcs, that is, arcs where the control is identically zero. Here we consider Pontryagin extremals given by a bang-inactive-bang concatenation. We establish sufficient optimality conditions for such extremals, in terms of some regularity conditions and of the coercivity of a suitable finite-dimensional second variation.
At the core of optimal control theory is the Pontryagin maximum principle - the celebrated first order necessary optimality condition - whose solutions are called extremals and which are obtained through a function called Hamiltonian, akin to the Lagrangian function used in ordinary calculus optimization problems. A remarkable property of the extremals is that the total derivative with respect to time of the corresponding Hamiltonian equals the partial derivative of the Hamiltonian with respect to time. In particular, when the Hamiltonian does not depend explicitly on time, the value of the Ha…
We prove necessary optimality conditions of Pontryagin type for a class of fuzzy fractional optimal control problems with the fuzzy fractional derivative described in the Caputo sense. The new results are illustrated by computing the extremals of three fuzzy optimal control systems, which improve recent results of Najariyan and Farahi.
The Pontryagin Maximum Principle for Nonlinear Infinite Horizon Optimal Control Problems with State Constraints
本文研究非线性无限时间最优控制问题，提出基于针形变分技术的方法，推导出强局部极小值的庞特里亚金最大值原理、Arrow型充分条件及多种横截条件。
In this paper we summarize our results in infinite horizon optimal control. We present optimality conditions for weak local minimizer in the framework of weighted functions. Moreover we formulate the Pontryagin Maximum Principle for strong local minimizer for trajectories converging at infinity. The considered problems, the requirements and the results depend on the choice of the function space.
Fixed-Time State Transfer via Pontryagin Extremals
研究非线性系统两状态间固定时间转移问题，提出通用公式，证明可达时解可由两个庞特里亚金极值拼接构造，等同于两点边值问题，还应用于三维曲线并给出数值演示。
In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on a manifold $M$ and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of an accessory problem on the tangent space to $M$ at the final point of the trajectory. Two examples are proposed.
In this paper we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k- dimensional closed ball. We prove the optimality of broken normal extremals when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the controllable fields is locally orthogonal to the singular locus and the drift does not belong to it. Moreover, if k = 2, we show the optimality of any broken extremal even abnormal when the controllable fields do not form a contact distribution in the poin…
In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is still open. We formulate the Reinhardt Conjecture as a problem in optimal control theory. The smoothed octagon is a Pontryagin extremal trajectory with bang-bang control. More generally, the smoothed regular $6k+2$-gon is a Pontryagin extremal with bang-bang control. The smoothed octagon is a strict (micro) local minimum to the optimal control problem. The optimal solution to the Reinhardt problem is a…
We consider a problem on maximizing the height of vertical flight of a material point ("meteorological rocket") in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The original Goddard problem is simplified by removing the dependence on the rocket mass from the equations of motion. Using the maximum principle we find all possible types of Pontryagin extremals and classify them w.r.t. problem parameters. Since the velocity of the point can be negative, we obtain some new types of extremals with two or three switching points, whi…
Pontryagin Maximum Principle for Rydberg-blockaded state-to-state transfers: A semi-analytic approach
本文研究了基于Rydberg阻塞机制的两量子比特和多量子比特操作的时间最优态到态控制问题，提出了一种半解析方法结合庞特里亚金最大原理，通过简化哈密顿量并分析极值路径，实现了高保真度的时间最优控制。
Second-order necessary conditions for optimal control problems are considered, where the ``second-order" is in the sense of that Pontryagin's maximum principle is viewed as a first-order necessary optimality condition. A sufficient condition for a local minimizer is also given.
In optimal control problems, there exist different kinds of extremals, that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic systems as long as they are equivalent. With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be …
Pontryagin maximum principle for non-smooth state-constrained control problems over Wasserstein spaces
本文针对 Wasserstein 空间上由非局部连续性方程支配的状态约束 Bolza 最优控制问题，提出了 Pontryagin 最大值原理，引入 Clarke 型次微分并利用 Lasry-Lions 正则化，为全局极小值提供一阶刻画。
This paper introduces a framework for solving time-autonomous nonlinear infinite horizon optimal control problems, under the assumption that all minimizers satisfy Pontryagin's necessary optimality conditions. In detail, we use methods from the field of symplectic geometry to analyze the eigenvalues of a Koopman operator that lifts Pontryagin's differential equation into a suitably defined infinite dimensional symplectic space. This has the advantage that methods from the field of spectral analysis can be used to characterize globally optimal control laws. A numerical method for constructing o…
The Pontryagin's Maximum Principle allows, in most cases, the design of optimal controls of affine nonlinear control systems by considering the sign of a smooth function. There are cases, although, where this function vanishes on a whole time interval and the Pontryagin's Maximum Principle alone does not give enough information to design the control. In these cases one considers the time derivatives of this function until a k-order derivative that explicitly depends on the control variable. The number q=k/2 is called problem order and it is the same to all the extremals. The local order is a r…
We consider the sub-Riemannian length minimization problem on the group of motions of pseudo Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin Maximum Principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates the vertical and the horizontal subsystems are integrat…
A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows is established using the Ekeland variational principle. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the linearized compressible Navier-Stokes equations. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.
While the Pontryagin Maximum Principle can be used to calculate candidate extremals for optimal orbital transfer problems, these candidates cannot be guaranteed to be at least locally optimal unless sufficient optimality conditions are satisfied. In this paper, through constructing a parameterized family of extremals around a reference extremal, some second-order necessary and sufficient conditions for the strong-local optimality of the free-time multi-burn fuel-optimal transfer are established under certain regularity assumptions. Moreover, the numerical procedure for computing these optimali…
This article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the so-called mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control a…
Consider, on the one part, a general nonlinear finite-dimensional optimal control problem and assume that it has a unique solution whose state is denoted by $x^*$. On the other part, consider the sampled-data control version of it. Under appropriate assumptions, we prove that the optimal state of the sampled-data problem converges uniformly to $x^*$ as the norm of the corresponding partition tends to zero. Moreover, applying the Pontryagin maximum principle to both problems, we prove that, if $x^*$ has a unique weak extremal lift with a costate $p$ that is normal, then the costate of the sampl…
This paper is a continuation of the work by the same authors on the Cartan group equipped with the sub-Finsler $\ell_\infty$ norm. We start by giving a detailed presentation of the structure of bang-bang extremal trajectories. Then we prove upper bounds on the number of switchings on bang-bang minimizers. We prove that any normal extremal is either bang-bang, or singular, or mixed. Consequently, we study mixed extremals. In particular, we prove that every two points can be connected by a piecewise smooth minimizer, and we give a uniform bound on the number of such pieces.
In the present paper we deal with fully nonlinear two-dimensional smooth control systems with scalar input $\dot{q} = \bs{f}(q,u)$, $q \in M$, $u \in U$, where $M$ and $U$ are differentiable smooth manifolds of respective dimensions two and one. For such systems, we provide two microlocal normal forms, i.e., local in the state-input space, using the fundamental necessary condition of optimality for optimal control problems: the Pontryagin Maximum Principle. One of these normal forms will be constructed around a regular extremal and the other one will be constructed around an abnormal extremal.…
A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. Apart from the design of this general algorithm useful for any optimal control problem, it is showed how it works to split the set of extremals and, in particular, to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.
Singular extremals of optimal control problems with $L^1$ cost
本文研究了具有 $L^1$ 成本的最优控制问题，提出奇极值的强广义勒杰兰-克勒布施条件，并证明其对于局部强最优性的必要性。
The total generalized variation (TGV) is a popular regularizer in inverse problems and imaging combining discontinuous solutions and higher order smoothing. In particular, empirical observations suggest that its order two version strongly favors piecewise affine functions. In the present manuscript, we formalize this statement for the one-dimensional TGV-functional by characterizing the extremal points of its sublevel sets with respect to a suitable quotient space topology. These results imply that 1D TGV-regularized linear inverse problems with finite dimensional observations admit piecewise …
Extremal Eigenvalues of Weighted Steklov Problems
研究边界密度函数对Steklov本征值的优化问题，分析极值本征值的存在性和结构特性，揭示最优解的 bang--bang 特性及非对称性，探讨凸优化工具的局限性及数值算法设计。
Helhmoltz-Kirchhoff equations of motions of vortices of an incompressible fluid in the plane define a dynamics with singularities and this leads to a Zermelo navigation problem describing the ship travel in such a field where the control is the heading angle. Considering one vortex, we define a time minimization problem which can be analyzed with the technics of geometric optimal control combined with numerical simulations, the geometric frame being the extension of Randers metrics in the punctured plane, with rotational symmetry. Candidates as minimizers are parameterized thanks to the Pontry…
We analyze an optimal control problem for systems of integral equations of Volterra type with two independent variables. These systems generalize both, the hyperbolic control problems for systems of Goursat-Darboux type, and the optimal control of ordinary (i.e. with one independent variable) Volterra integral equations. We prove extremal principles akin to Pontryagin's maximum principle.
We study, in a unified way, the following questions related to the properties of Pontryagin extremals for optimal control problems with unrestricted controls: i) How the transformations, which define the equivalence of two problems, transform the extremals? ii) How to obtain quantities which are conserved along any extremal? iii) How to assure that the set of extremals include the minimizers predicted by the existence theory? These questions are connected to: i) the Caratheodory method which establishes a correspondence between the minimizing curves of equivalent problems; ii) the interplay be…
We obtain a generalization of Noether's invariance principle for optimal control problems with equality and inequality state-input constraints. The result relates the invariance properties of the problems with the existence of conserved quantities along the constrained Pontryagin extremals. A result of this kind was posed as an open question by Vladimir Tikhomirov, in 1986.
This paper addresses the time-optimal control problem for a class of control systems which includes controlled mechanical systems with possible dissipation terms. The Lie algebras associated with such mechanical systems enjoy certain special properties. These properties are explored and are used in conjunction with the Pontryagin maximum principle to determine the structure of singular extremals and, in particular, time-optimal trajectories. The theory is illustrated with an application to a time-optimal problem for a class of underwater vehicles
Extremal points and sparse optimization for generalized Kantorovich-Rubinstein norms
本文刻画了广义Kantorovich-Rubinstein范数下水平集的极值点结构，推导了一阶最优性条件，提出带线性收敛保证的高效求解算法，数值实验验证了其收敛特性。
We study a time minimization problem on the group of motions of a plane with admissible control in a half-disk. The considered control system describes a model of a car that can move forward on a plane and turn in place. Optimal trajectories of this system are used in image processing for the detection of salient lines. In particular, such trajectories are used in the analysis of medical images when searching for blood vessels in photos of the human retina. The problem is of interest in geometric control theory as a model example in which the set of admissible controls contains zero at the bou…
Let $T>0$ fixed. We consider the optimal control problem for analytic affine systems: $\ds{\dot{x}=f\_0(x)+\sum\_{i=1}^m u\_if\_i(x)}$, with a cost of the form: $\ds{C(u)=\int\_0^T \sum\_{i=1}^m u\_i^2(t)dt}$. For this kind of systems we prove that if there are no minimizing abnormal extremals then the value function $S$ is subanalytic. Secondly we prove that if there exists an abnormal minimizer of corank 1 then the set of end-points of minimizers at cost fixed is tangent to a given hyperplane. We illustrate this situation in sub-Riemannian geometry.
The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics. The main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive description of different zones of interest using viability tools. The second achievement is a thorough mathematical analysis of Pontryagin extremals for the aforementioned problem allowing to obtain an explicit closed-loop feedback optimal control. A…
We discuss contact geometry naturally related with optimal control problems (and Pontryagin Maximum Principle). We explore and expand the observations of [Ohsawa, 2015], providing simple and elegant characterizations of normal and abnormal sub-Riemannian extremals.
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnor…
Normality and Gap Phenomena in Optimal Unbounded Control
本文研究最优无界控制问题中从原始问题过渡到扩展脉冲版本是否引入下确界间隙，基于极值的正规性提供了不存在该间隙的充分条件，并在某些情况下简化为可验证准则以改进先前结果。
Optimal control problems are usually addressed with the help of the famous Pontryagin Maximum Principle (PMP) which gives a generalization of the classical Euler-Lagrange and Weierstrass necessary optimality conditions of the calculus of variations. Success in applying the PMP permits to obtain candidates for a local minimum. In 1967 a direct method, which permits to obtain global minimizers directly, without using necessary conditions, was introduced by Leitmann. Leitmann's approach is connected, as showed by Carlson in 2002, with "Caratheodory's royal road of the Calculus of variations". Her…
KLAP: KYP lemma based low-rank approximation for $\mathcal{H}_2$-optimal passivation
本文提出基于KYP引理和Lur'e方程的无源化方法KLAP，通过求解无约束优化问题寻找最小化 $\mathcal{H}_2$ 距离的扰动，并证明了局部极小值的全局最优性，提出了高效的初始化与重启策略。
Universal Birkhoff Method for Computing Extremals in the Elliptic Restricted Three-Body Problem
该研究针对椭圆型限制性三体问题轨迹优化，结合通用伯克霍夫理论与快速无猜测谱算法，实现无需动力学系统辅助的可验证极值计算。
We have found an inconsistency in our previous version of the paper "Generalized splines in R^n and optimal control". We give a new-time-dependent definition of spline curves in R^n which results from solving a non-autonomous linear quadratic optimal control problem (P) where the matrix B(t) is assumed to be rectangular with maximum rank. Nevertheless, our results are only valid if B(t) is a square (nonsingular) matrix. This was pointed out to us by Andrey Sarychev. We have proceeded with the necessary corrections. %%%%%%%%%%%%%%%%%% We give a new time-dependent definition of spline curves in …
Optimal control for coupled sweeping processes under minimal assumptions
本文在最少数假设下研究了耦合扫掠过程的最优控制问题，推导了非光滑Pontryagin极大原理，并建立了动态方程的Lipschitz解的存在性，同时提出了改进的指数惩罚近似技术。
In optimal control theory, infimum gap means that there is a gap between the infimum values of a given minimum problem and an extended problem, obtained by enlarging the set of original solutions and controls. The gap phenomenon is somewhat "dual" to the problem of the controllability of the original control system to an extended solution. In this paper we present sufficient conditions for the absence of an infimum gap and for controllability for a wide class of optimal control problems subject to endpoint and state constraints. These conditions are based on a nondegenerate version of the nons…
We describe a reformulation (following Hales (2017)) of a 1934 conjecture of Reinhardt on pessimal packings of convex domains in the plane as a problem in optimal control theory. Several structural results of this problem including its Hamiltonian structure and Lax pair formalism are presented. General solutions of this problem for constant control are presented and are used to prove that the Pontryagin extremals of the control problem are constrained to lie in a compact domain of the state space. We further describe the structure of the control problem near its singular locus, and prove that …
In this paper, we investigate the minimal time problem for the guidance of a rocket, whose motion is described by its attitude kinematics and dynamics but also by its orbit dynamics. Our approach is based on a refined geometric study of the extremals coming from the application of the Pontryagin maximum principle. Our analysis reveals the existence of singular arcs of higher-order in the optimal synthesis, causing the occurrence of a chattering phenomenon, i.e., of an infinite number of switchings when trying to connect bang arcs with a singular arc. We establish a general result for bi-input …
We study in optimal control the important relation between invariance of the problem under a family of transformations, and the existence of preserved quantities along the Pontryagin extremals. Several extensions of Noether theorem are provided, in the direction which enlarges the scope of its application. We formulate a more general version of Noether's theorem for optimal control problems, which incorporates the possibility to consider a family of transformations depending on several parameters and, what is more important, to deal with quasi-invariant and not necessarily invariant optimal co…
In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously…
In this article, the minimum time control problem of an electric vehicle is modeled as a Mayer problem in optimal control, with affine dynamics with respect to the control and with state constraints. The candidates as minimizers are selected among a set of extremals, solutions of a Hamiltonian system given by the maximum principle. An analysis, with the techniques of geometric control, is used first to reduce the set of candidates and then to construct the numerical methods. This leads to a numerical investigation based on indirect methods using the HamPath software. Multiple shooting and homo…
Exponential Turnpike Theorems for Nonlinear Deterministic Meanfield Optimal Control Problems
本文在拉格朗日与欧拉框架下，建立非线性确定性均场最优控制问题的指数趋中定理，通过一阶最优条件、二阶哈密顿展开和算子里卡蒂对角化论证，证明了在希尔伯特空间中的指数趋中定理，并探讨了Wasserstein空间中的最优庞特里亚金三元组。
This article contributes to a framework for a computational indirect method based on the Pontryagin maximum principle to efficiently solve a class of state constrained time-optimal control problems in the presence of a time-dependent flow field. Path-planning in a flow with tidal variations is an emblematic real-life task that serves as an example falling in the class of problems considered here. Under rather natural assumptions on the flow field, the problem is regular and the measure Lagrange multiplier, associated with the state constraint, is continuous. These properties (regularity and co…
Optimal Control of ensembles of dynamical systems
本文针对仿射控制系统集合的最优控制问题，通过$\Gamma$-收敛用有限子集合逼近无限集合，推导出极大值原理，并提出基于梯度投影和庞特里亚金原理的数值算法，在二维线性系统上验证了有效性。
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