Global and Local Adjoint Formulations for (GOGP)

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Standard Results on Hamiltonian Fields

We denote by ⁄k (M) the vector bundle of all k-forms on M. A differential k-form (or simply, k-form, when no misunderstanding arises) is a section of ⁄k (M), i.e. a smooth function · : M ! ⁄k (M) such that ·q ˘ ·(q) 2 ⁄kq (M). The space of all k-forms is denoted by Ak (M). Let N be a smooth manifold and F : M ! N be a smooth mapping. We call the pull-back of · 2 Ak (N ) the k-form F ⁄· 2 Ak (M) obtained by (F ⁄·)q (v1, . . . , vk ) ˘ ·F (q)(d F (q)(v1), . . . ,d F (q)(vk )) for every q 2 M and every v1, . . . , vk 2 Tq M.
The set A1(M) is the space of all 1-forms on M, i.e. smooth mappings s : M ! T ⁄M such that s(q) 2 Tq⁄M. We recall that any local chart (V,’) of M determines canonical local coordinates of the form (x,») ˘ (x1, . . . , xn ;»1, . . . ,»n ) for the cotangent bundle T ⁄M. Then, any covector p 2 Tq⁄M has the decomposition p ˘ Pni˘1 »i d xi jq , where we denote d xi ˘ (@xi )⁄ 2 A1(M).
The cotangent bundle T ⁄M can be equipped with a canonical symplectic structure as follows. Consider the projection … : T ⁄M ! M and define the Liouville 1-form as s : T ⁄M ! T ⁄(T ⁄M) : p …⁄p 2 T ⁄(T ⁄M) . 7! p In local coordinates, the 1-form s can be written as s(p) ˘ in˘1 »i d xi j…(p). Then, by taking the differential of s, we obtain the nondegenerate closed 2-form ¾ ˘ d s ˘ P Pn d»i ^d xi 2 A2(T ⁄M), which makes (T ⁄M,¾) a symplectic manifold. i ˘1.
We call Hamiltonian every arbitrary smooth function on the cotangent bundle. To any Hamiltonian h 2 C 1(T ⁄M), we can associate a unique Hamiltonian vector field h : T ⁄M ! T (T ⁄M) such that ¾(p)(¢,h) ˘ d h(p) for every p 2 T ⁄M. In canonical co- ‡ · ordinates, one has h ˘ ni˘1 @»@hi @xi ¡ @@xhi @»i . We call Hamiltonian dynamical prob- the following dynamical problem related to the field h lem corresponding to hP p˙(t) ˘ h(p(t)) , p(t0) ˘ p0 2 T…⁄(p0)M.

Optimality Conditions and Numerical Methods

More informative relations on optimal controls related to (OCP) can be achieved by refining the Lagrange multiplier rule, arising conditions like the Maximum Principle, whose aim consists in reformulating optimal control problems into initializations of ordinary differential equation systems. A detailed analysis of optimal arcs leads also to sufficient conditions of optimality. All this information can be exploited to set up several numerical resolution strategies for (OCP), as we develop hereafter.

Sufficient Optimality Conditions

Since the Maximum Principle is a necessary condition, an extremal fulfilling the con-ditions of Theorem 1.1 may not be related to an optimal solution of (OCP). However, under appropriate assumptions, further conditions exist to ensure the optimality of normal extremals. We recall here some basic facts concerning sufficient optimality conditions, used in Chapter 3 of this thesis (we refer to [34, Chapter 17]).
Suppose that M is simply connected and consider the following autonomous version of (OCP), for which we fix a final time t f and we minimize the cost C (u) ˘ Z0t f f 0(q(t),u(t)) d t such that ˙ q(0) ˘ q0 , q(t f ) ˘ q f 2 M f q(t) ˘ f (q(t),u(t)) , among all the controls u 2 L1([0,T ],Rm ) satisfying u(t) 2U a.e. in [0, t f ], where q f is a fixed final state. The Hamiltonian corresponding to normal extremals is h(q, p,u) ˘ hu (p) ˘ hp, f (q,u)i ¡ f 0(q,u) , q ˘ …(p) , p 2 Tq⁄M , u 2 Rm .

Numerical Difficulties Due to Control and State Constraints

Even if the adjustments to run direct methods on (OCP)m,s and (OCP)m are provided straightforwardly by discretizing the constraints depending on the state, by analyzing both Theorem 1.3 and Theorem 1.4, it is easily understood that adapting numerical methods such as shooting or multi-shooting algorithms becomes complicated when considering control and state constraints. Indeed, even if we assume to deal with regular controls, the presence of the multipliers „me , „mi and „s prevents from inte-grating the adjoint equations. As pointed out previously, this is due to the fact that, usually, no knowledge concerning the evolution of these multipliers is provided. Obtaining rigorous and useful information on the evolution and the regularity of „me , „mi and „ s may be arduous and has been the object of many studies in the existing literature, both from theoretical (for example, high-order analysis [19, 21, 71]) and numerical point of views (for example, aerospace applications [4, 26]). Even if indi-rect methods to solve (OCP)m,s and (OCP)m have already been proposed, they are able to work under very particular assumptions and make the numerical compu-tations more demanding, often losing the fast convergence of the original shooting method (for further details on these procedures, we refer to [20, 72, 73, 74]).

Problems with Control and State Delays

We conclude this chapter by introducing necessary optimality conditions and dif-ficulties of related indirect methods for optimal control problems with control and state delays. The interest of introducing this kind of problems arises from the fact that, in the context of our launch vehicle application, delays coming from model re-finement often occur. Therefore, related to our main challenge of solely exploiting indirect methods, we need to understand whether it is possible to efficiently solve optimal control problems with control and state delays by shooting-type procedures.

Maximum Principle for Problems with Delays

For matter of concision, in what follows, all the concerned results are developed con-sidering systems evolving in the Euclidean space, i.e. M ˘ Rn , and subject to pure control constraints. This is not limiting to address our launch vehicle application. A nonautonomous vector field with delays on Rn is a continuous vector function f : R£R2n £R2m ! Rn : (t, x, y,u, v) 7!f (t, x, y,u, v) which is smooth w.r.t. its second and third variables. Without loss of generality, we assume that any considered vector field with delays has a compact support. Fix a positive value ¢ and consider an initial state function ‘1 2 C 0([¡¢, 0],Rn ) as well as an initial control function ‘2 2 L1([¡¢, 0),Rm ). For every couple of delays ¿ ˘ (¿1,¿2) 2 [0,¢]2 and every control u 2 L1loc ([¡¢,1),Rm ) such that uj[¡¢,0)(¢) ˘ ‘2(¢), the following dynamical system with delays x˙(t) ˘ f (t, x(t), x(t ¡¿1),u(t),u(t ¡¿2)) , xj[¡¢,0](¢) ˘ ‘1(¢) (1.29) is well-defined. As a classical result, there exists a unique curve x(¢) defined for ev-ery t 2 [¡¢, 1) and satisfying (1.29), which depends continuously (w.r.t. appropriate topologies) on the initial data ¿ 2 [0,¢]2, u 2 L1loc ([¡¢,1),Rm ), ‘1 2 L1([¡¢, 0],Rn ). Consider a continuous integral cost function with delays f 0 : R£R2n £R2m ! R : (t, x, y,u, v) 7!f 0(t, x, y,u, v).

Numerical Difficulties Due to Control and State Delays

From the arguments presented in Section 1.3.3, it is clear that adapting direct meth-ods to solve (OCP)¿ does not produce any obstacle. However, solving (OCP)¿ from a numerical point of view by means of indirect methods becomes complex because a global information on the adjoint vector p is needed. This is explained as follows.
Assume that the optimal control u(¢) is regular, i.e. it can be written as a function of x(¢) and p(¢) (by the Maximality Condition, see Section 1.3.1). Therefore, each itera-tion of a shooting method consists in solving the coupled dynamics coming from the adjoint equations, where a value of p(0) is provided. In the context of Theorem 1.5, this means that one has to solve Differential-Difference Boundary Value Problems (DDBVP), where both forward and backward terms of time appear within mixed type differential equations. The difficulty in solving DDBVP is the lack of global informa-tion which forbids a purely local integration by usual iterative methods for ordinary differential equations. Techniques to solve mixed type differential equations aris-ing from DDBVP have been analyzed, such as analytical decompositions of solutions [81, 82], and related numerical schemes [83, 84]. In these approaches, the dimension of the problem may drastically increase as much as the numerical accuracy increases. The previous considerations show that, in order to initialize correctly a shooting method on (OCP)¿, a standard guess of the initial value of the adjoint vector p(0) is not sufficient, but rather, a good numerical guess of the whole function p(¢) must be provided to make the procedure converge. This represents an additional difficulty with respect to the usual shooting method and it requires a global discretization of the adjoint equations, increasing considerably the dimension of the problem.

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Environmental and Dynamical Modeling

By assumption, we model the launch vehicle as an axial symmetric rigid body of mass m. Its motion is described by the evolution of the state variables (r,v,b,ω), where r ˘ xI ¯yJ ¯zK is the trajectory of its center of gravity G, v ˘ x˙I ¯y˙J ¯z˙K denotes its velocity and the normal vector b corresponds to the principal body axis, whose time evolution is defined by the angular velocity ω. We proceed firstly by analyzing the environment, and thereafter, by providing the models of physical quantities. All the concerned aerodynamical forces depend on the interaction between the at-mosphere and the vehicle. It is crucial to provide a model of the air density. We present a concise introduction and further details on atmospheric models can be found in [8, 85, 86, 87]. In the following, we define the altitude as the scalar quantity h ˘ kr ¡rT k, where rT ˘ krT k ˘ 6378, 145 km is the radius of the Earth.

Optimal Interception Problem (OIP)

From a numerical viewpoint, we focus on a particular subclass of (GOGP): the endo-atmospheric interception. The context can be summarized as follows (see, e.g. [6]). The target is represented by a supersonic missile whose position is assumed to be known at each time step. The objective consists in intercepting the target employing a ground-to-air/air-to-air missile by maximizing the chances to neutralize the threat. Many phases occur to control the vehicle. During the first phase, when the intercept-ing missile is launched, some predefined controllers stabilize its critical movements due to its too low velocity. When a certain threshold value of the velocity is reached, the mid-course phase controllers start to guide the vehicle; this phase is the longest one and needs a control strategy able to provide the best conditions to intercept the target when the interceptor reach some precomputed impact point. When this point is attained, the mid-course controller stops allowing the last, usually automatized, control strategy to guide the intercepting missile to the impact with the threat.

Optimal Interception Problem with Delays (OIP)¿

The last problem that we consider is a variant of the optimal interception problem, in which a control on the rotational velocity is introduced, considering moreover some delay on mechanical information communication. This problem allows to simulate phenomena like the non-minimum phase problem, whose principle is as follows (see, e.g. [95, 96]). Consider a bank-to-turn or a skid-to-turn vehicle (such as the dynamical system considered for (OIP), see, e.g. [8, 6]). In order to gain altitude, the first typical maneuver executed by the control system consists in rotating the air-craft to increase the angle of attack. Rotating the vehicle produces a temporary loss of pressure that develops a downward force at the tail. This causes an overall downward force on the aircraft that initially lowers the center of gravity, before the increased up-ward force on the main wings from the increased angle of attack raises the vehicle. Practically, this phenomenon can be easily reproduced by inserting a delay between the variation of the angular velocity and the effect of the lift raising the vehicle.

Table of contents :

Introduction Générale
General Introduction
I Optimal Control Framework and Dynamic alModel 
1 Elements of Optimal Control 
1.1 Some Tools from Differential Geometry
1.1.1 Notations and Properties of Vector Fields
1.1.2 Standard Results on Hamiltonian Fields
1.2 Classical Optimal Control Problems
1.3 Optimality Conditions and NumericalMethods
1.3.1 TheMaximumPrinciple
1.3.2 Sufficient Optimality Conditions
1.3.3 Classical NumericalMethods in Optimal Control
1.3.4 Numerical HomotopyMethods
1.4 Problems with Control and State Constraints
1.4.1 General Control and State Constraints
1.4.2 Mixed Control-State Constraints
1.4.3 Numerical Difficulties Due to Control and State Constraints
1.5 Problems with Control and State Delays
1.5.1 MaximumPrinciple for Problems with Delays
1.5.2 Numerical Difficulties Due to Control and State Delays
2 Rendezvous Problems 
2.1 Physical Problem and DynamicalModel
2.1.1 Fundamental Coordinate Systems
2.1.2 Environmental and DynamicalModeling
2.2 Optimal Control Problems
2.2.1 General Optimal Guidance Problem (GOGP)
2.2.2 Optimal Interception Problem (OIP)
2.2.3 Optimal Interception Problem with Delays (OIP)¿
II Structure of Extremals and Numerical Strategies of Guidance 
3 Structure of Extremals for Optimal Guidance Problems 
3.1 Local Change of Problems Under Abstract Framework
3.1.1 Reduction to Local Problems with Pure Control Constraints
3.1.2 Sufficient Conditions Under Reduction to Local Problems
3.2 Local Transformations for (GOGP)
3.2.1 Coordinates Under the Trajectory Reference Frame
3.2.2 Additional Local Euler Coordinates
3.2.3 Global and Local Adjoint Formulations for (GOGP)
3.3 Regular and Nonregular Pontryagin Extremals
3.3.1 Regular Pontryagin Extremals
3.3.2 Nonregular Pontryagin Extremals
3.4 Conclusions
4 Numerical Guidance Strategy 
4.1 General Numerical Homotopy Procedure for (GOGP)
4.1.1 General Optimal Guidance Problem of Order Zero (GOGP)0
4.1.2 Parametrized Family of Optimal Control Problems (GOGP)
4.2 Optimal Interception Problem of Order Zero (OIP)0
4.2.1 Approximated Local Controllability of (OIP)s0
4.2.2 Froma LOS Analysis to a Suboptimal Guidance Law for (OIP)s0
4.3 Numerical Simulations for (OIP)
4.3.1 Mathematical Design of theMission
4.3.2 Homotopy Scheme and Numerical Results
4.4 Conclusions
5 Numerical Robustness and Interception Software (ONERA) 
5.1 Increasing the Robustness: Initialization Grids
5.1.1 Fast Initialization Grids Design
5.1.2 Numerical Time-RobustnessMonte Carlo Experiments
5.2 Software Design: a Template C++ Library (ONERA)
5.2.1 Library Structure (Simplified UML Class Diagram)
5.2.2 Details on Classes and User Script Examples
5.3 Conclusions
III Continuity of Pontryagin Extremals with Respect to Delays 
6 Solving Optimal Control Problems with Delays 
6.1 Continuity Properties with Respect to Delays
6.2 Homotopy Algorithmand Numerical Simulations
6.2.1 Solving (OCP)¿ by ShootingMethods and Homotopy on Delays
6.2.2 First Numerical Tests
6.3 Numerical Strategy to Solve (OIP)¿
6.3.1 Local Initialization Procedure for (OIP)¿
6.3.2 Numerical Simulations for (OIP)¿
6.4 Conclusions
7 Continuity Properties of Pontryagin Extremals 
7.1 Proof of the PMP Using Needle-Like Variations
7.1.1 Preliminary Notations
7.1.2 Needle-Like Variations and Pontryagin Cones
7.1.3 Proof of TheMaximum Principle
7.2 Conic Implicit Function Theorem with Parameters
7.3 Proof of Theorem 6.1
7.3.1 Controllability for (OCP)¿
7.3.2 Existence of Optimal Controls for (OCP)¿
7.3.3 Convergence of Optimal Controls and Trajectories for (OCP)¿ .
7.3.4 Convergence of Optimal Adjoint Vectors for (OCP)¿
7.4 Conclusions


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