Direct Method for Resolution of Optimal Control Problem with Free Initial Condition

Abstract

The theory of control analyzes the proprieties of commanded systems. Problems of optimal control (OC) have been intensively investigated in the world literature for over forty years. During this period, series of fundamental results have been obtained, among which should be noted the maximum principle (Pontryagin et al., 1962) and dynamic programming (Bellman, 1963). For many of the problems of the optimal control theory (OCT), adequate solutions are found (Bryson and Yu-chi, 1969, Lee and Markus, 1967, Gabasov and Kirillova, 1977, 1978, 1980). Results of the theory were taken up in various fields of science, engineering, and economics. The present paper aims at extending the constructive methods of Balashevich et al., (2000) that were developed for the problems of optimal control with the bounded initial state is not fixed are considered.

1. Introduction

Problems of optimal control (OC) have been intensively investigated in the world literature for over forty years. During this period, a series of fundamental results have been obtained, whose majority is based on the maximum principle [1] and dynamic programming [2–4]. Currently there exist two types of methods of resolution: direct methods and indirect methods. The indirect methods are based on the maximum principle [1] and the methods of the shooting [5]. The direct methods are based on the discretization of the initial problem, but here we obtain an approximate solution.

The aim of this paper is to apply an adaptive method of linear programming [6–13] for an optimal control problem with a free initial condition. Here we use a final procedure based on a resolution of linear system with the Newton method to obtain a optimal solution. Here, we use a finite set of switching points of a control [11, 14–16]. We solve the same problem in the article [17], we transform a problem initial to a problem of linear programming by carrying changes of variables in three procedures: change of control, change of support, and final procedure, in our paper, a solution of this problem, we discretize a problem initial to find an optimal support by using change of control and change of support, and we present the final procedure which uses this solution as an initial approximation for solving problem in the class of piecewise continuous function.

We explain below that the realizations of the adaptive method [18] described in the paper possess the following advantages.(1)Size of the support (the main tool of the method), which mainly influences the complexity of an iteration of the method, does not depend on all general constraints but only on the quantity of endpoint constraints.(2)In operations of described realizations, only parameters of the initial control problem are used. This consideration decreases requirements to operative memory and increases accuracy of calculations.(3)Main operations are conducted with initial (primal) and adjoint systems without auxiliary objects arising after reduction of the initial optimal control problem to the equivalent LP problem.(4)Because of storing a little volume of additional information and using parallel calculations, the time for integration of primal and adjoint systems in the dual part of an iteration decreases substantially. This precipitates the solution to the open-loop optimization problem and also the formation of current supports and realizations of optimal feedbacks when positional solutions are constructed.(5)Effectiveness of methods is practically independent of a quantization period.

The paper has the following structure: in Section 2, the canonical optimal control problem is formulated and the definition of support is introduced. Primal and dual ways of its dynamical identification are given. In Section 3, optimality and suboptimality criterion are given. In Section 4, optimality and 𝜀-optimality criteria are exposed. In Section 5, numerical algorithm for solving the problem is discussed. The iteration consists in three procedures: change of control, change of a support, and at the end, final procedure. In Section 6, the results are illustrated with a numerical example.

2. Statement of the Problem

On the time interval 𝑇=[0,𝑡^∗], we have the following linear problem of optimal control:𝐽(𝑧,𝑢(𝑡))=𝑐^′𝑥𝑡^∗(0)⟶max,𝑥̇=𝐴𝑥+𝑏𝑢,𝑥=𝑧∈𝑋₀=𝑧∈ℜ^𝑛,𝐺𝑧=𝛾,𝑑_∗≤𝑧≤𝑑^∗𝑡,𝐻𝑥^∗𝑓=𝑔,_∗≤𝑢(𝑡)≤𝑓^∗,𝑡∈𝑇=0,𝑡^∗. (2.1)(2.2)(2.3)(2.4) Here 𝑥∈ℜ^𝑛 is a state of control system (2.2); 𝑢(⋅)=(𝑢(𝑡),𝑡∈𝑡), 𝑇=[0,𝑡^∗], is a piecewise continuous function; 𝐴∈ℜ^𝑛×𝑛; 𝑏,𝑐∈ℜ^𝑛; 𝑔∈ℜ^𝑚×𝑛, rank𝐻=𝑚≤𝑛; 𝑓_∗,𝑓^∗ are scalars; 𝑑_∗=(𝑑_∗𝑗,𝑗∈𝐽), 𝑑^∗=𝑑^∗(𝐽)=(𝑑^∗_𝑗,𝑗∈𝐽) are 𝑛-vectors; 𝐺∈ℜ^𝑙×𝑛, rank𝐺=𝑙≤𝑛,𝛾∈ℜ^𝑙, 𝐼={1,…,𝑚}, 𝐽={1,…,𝑛}, 𝐿={1,…,𝑙} are sets of indices.

By using the Cauchy formula, we obtain the solution of system (2.2):𝑥(𝑡)=𝐹(𝑡)𝑧+^𝑡₀𝐹⁻¹(𝜗)𝑏𝑢(𝜗)𝑑𝜗,𝑡∈𝑇, (2.5) where 𝐹(𝑡)=𝑒^𝐴𝑡, 𝑡∈𝑇=[0,𝑡^∗], is the solution of the system:𝐹̇(𝑡)(𝑡),𝐹(0)=𝐴𝐹=𝐼_𝑛. (2.6) By using the formula (2.5) for 𝑡^∗, problem (2.1)–(2.4) becomes the equivalent following problem:𝑐^′̃𝑧+^𝑡∗₀𝑐(𝑡)𝑢(𝑡)𝐷()𝑑𝑡⟶max,𝐼,𝐽𝑧+^𝑡∗₀𝜑(𝑡)𝑢(𝑡)𝐺()𝑑𝑡=𝑔,𝐿,𝐽𝑧=𝛾,𝑑_∗≤𝑧≤𝑑^∗𝑓_∗,≤𝑢(𝑡)≤𝑓^∗,𝑡∈𝑇, (2.7)(2.8)(2.9)(2.10) where 𝑐^′̃=𝑐^′𝐹(𝑡^∗), 𝑐(𝑡)=𝑐^′𝐹(𝑡^∗)𝐹⁻¹(𝑡)𝑏, 𝐷(𝐼,𝐽)=𝐻𝐹(𝑡^∗), and 𝜑(𝑡)=𝐻𝐹(𝑡^∗)𝐹⁻¹(𝑡)𝑏.

3. Fundamental Definitions

Definition 3.1. A pair 𝑣=(𝑧,𝑢(⋅)) formed of an 𝑛-vector 𝑧 and a piecewise continuous function 𝑢(⋅) is called a generalized control.

Definition 3.2. A generalized control 𝑣=(𝑧,𝑢(⋅)) is said to be an admissible control if it satisfies the constraints (2.2)–(2.4).

By using this notation, a functional becomes𝐽(𝑧,𝑢(𝑡))=𝐽(𝑣)=𝑐^′𝑥𝑡^∗=𝑐^′̃𝑧+^𝑡∗₀𝑐(𝑡)𝑢(𝑡)𝑑𝑡. (3.1)

Definition 3.3. An admissible control 𝑣⁰=(𝑧⁰,𝑢⁰(⋅)) is said to be an optimal open-loop control if a control criterion reaches its maximal value 𝐽𝑣⁰=max_𝑣𝐽(𝑣). (3.2)

Definition 3.4. For a given 𝜀≥0, a control 𝑣^𝜀=(𝑧^𝜀,𝑢^𝜀(⋅)) is said to be 𝜀-optimal (approximate solution) if 𝐽𝑣⁰−𝐽𝑣^𝜀≤𝜀. (3.3)

4. Support Control

In the interval 𝑇, let us choose subset 𝑇_ℎ={0,ℎ,…,𝑡^∗−ℎ} formed of an isolated moment, where ℎ=𝑡^∗/𝑁, 𝑁 is an integer. A function 𝑢(𝑡), 𝑡∈𝑇, is called a discrete control if𝑢(𝑡)=𝑢(𝜏),𝑡∈[𝜏,𝜏+ℎ),𝜏∈𝑇_ℎ. (4.1) By using this discretization, problem (2.7)–(2.10) becomes𝑐′̃𝑧+_{𝑡∈𝑇ℎ}𝑞(𝑡)𝑢(𝑡)𝐷()⟶max,𝐼,𝐽𝑧+_{𝑡∈𝑇ℎ}𝑑(𝑡)𝑢(𝑡)𝐺()=𝑔,𝐿,𝐽𝑧=𝛾,𝑑_∗≤𝑧≤𝑑^∗𝑓_∗,≤𝑢(𝑡)≤𝑓^∗,𝑡∈𝑇. (4.2)(4.3)(4.4)(4.5)𝑑(𝑡) are defined by the following expression:𝑑(𝑡)=_𝑡^𝑡+ℎ𝜑(𝜗)𝑑𝜗=_𝑡^𝑡+ℎ𝜓^′(𝜗)𝑏(𝜗)𝑑𝜗, (4.6) and 𝑞(𝑡) equal𝑞(𝑡)=_𝑡^𝑡+ℎ𝑐(𝜗)𝑑𝜗=_𝑡^𝑡+ℎ𝜚(𝜗)𝑑𝜗,𝑡∈𝑇_ℎ. (4.7) Here 𝜓_𝑐(𝑡),𝑡∈𝑇, are a solution to the dual equation𝜓̇=−𝐴′𝜓, (4.8) with the initial condition𝜓𝑡⁰=𝑐, (4.9) and 𝜚(𝑡), 𝑡∈𝑇, is an 𝑚×𝑛 matrix function solution of the following equation:𝜚̇=−𝜚𝐴, (4.10) with the initial condition𝜚𝑡⁰=𝐻. (4.11)

First we solve problem (4.2)–(4.5); we construct the support: choose an arbitrary subset 𝑇_𝐵⊂𝑇_ℎ of 𝑘≤𝑚 elements and an arbitrary subset 𝐽_𝐵⊂𝐽 of 𝑚+𝑙−𝑘 elements. Form the matrix𝑃_𝐵=𝐷𝐼,𝐽_𝐵𝑑(𝑡),𝑡∈𝑇_𝐵𝐺𝐿,𝐽_𝐵0. (4.12)

A set 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} is said to be a support of problem (2.1)–(2.4) if det𝑃_𝐵≠0.

A pair {𝑣,𝑆_𝐵} of an admissible control 𝑣=(𝑧,𝑢(⋅)) and a support 𝑆_𝐵 is said to be a support control. A support control {𝑣,𝑆_𝐵} is said to be not degenerate if 𝑑_∗𝑗<𝑧_𝑗<𝑑^∗_𝑗, 𝑗∈𝐽_𝐵, 𝑓_∗<𝑢(𝑡)<𝑓^∗, 𝑡∈𝑇_𝐵.

Let us consider another admissible control 𝑣=(𝑧,𝑢(⋅))=𝑣+Δ𝑣, where 𝑧=𝑧+Δ𝑧, 𝑢(𝑡)=𝑢(𝑡)+Δ𝑢(𝑡), 𝑡∈𝑇, and let us calculate the increment of the cost functional:Δ𝐽(𝑣)=𝐽(𝑣)−𝐽(𝑣)=𝑐′̃Δ𝑧+_{𝑡∈𝑇ℎ}𝑞(𝑡)Δ𝑢(𝑡). (4.13) As 𝑣 is admissible, then we have𝐷(𝐼,𝐽)Δ𝑧+_{𝑡∈𝑇ℎ}𝑑(𝑡)(𝑡)𝐺()Δ𝑢=0,𝐿,𝐽Δ𝑧=0, (4.14) and consequently the increment of the functional is equal toΔ𝐽(𝑣)=𝑐^′̃−𝜈^′𝐷()𝐺()𝐼,𝐽𝐿,𝐽Δ𝑧+_{𝑡∈𝑇ℎ}𝑞(𝑡)−𝜈^′𝑑(𝑡)Δ𝑢(𝑡), (4.15) where 𝜈=^𝜈𝑢_𝜈𝑧∈𝑅^𝑚+𝑙, 𝜈_𝑢∈𝑅^𝑚, 𝜈_𝑧∈𝑅^𝑙, is a function of the Lagrange multipliers called potentials, calculated as a solution to the equation 𝜈′=𝑞′_𝐵𝑄, where 𝑄=𝑃_𝐵⁻¹, 𝑞_𝐵=(𝑐_𝑗̃,𝑗∈𝐽_𝐵,𝑞(𝑡),𝑡∈𝑇_𝐵). Introduce an 𝑛-vector of estimates Δ^′=𝜈^{′𝐷(𝐼,𝐽)}_{𝐺(𝐿,𝐽)}−𝑐′̃, and a function of cocontrol Δ(⋅)=(Δ(𝑡)=𝜈^′_𝑢𝑑(𝑡)−𝑞(𝑡),𝑡∈𝑇_ℎ).

By using this vector, the cost of functional increment takes the formΔ𝐽(𝑣)=Δ^′Δ𝑧−_{𝑡∈𝑇ℎ}Δ(𝑡)Δ𝑢(𝑡). (4.16) A support control {𝑣,𝑆_𝐵} is dually not degenerate if Δ(𝑡)≠0, 𝑡∈𝑇_𝐻, Δ_𝑗≠0, 𝑗∈𝐽_𝐻, where 𝑇_𝐻=𝑇_ℎ/𝑇_𝐵, 𝐽_𝐻=𝐽/𝐽_𝐵.

5. Calculation of the Value of Suboptimality

The new control 𝑣(𝑡) is admissible if it satisfies the constraints:𝑑_∗−𝑧≤Δ𝑧≤𝑑^∗−𝑧;𝑓_∗−𝑢(𝑡)≤Δ𝑢(𝑡)≤𝑓^∗−𝑢(𝑡),𝑡∈𝑇. (5.1) The maximum of functional (4.16) under constraints (5.1) is reached forΔ𝑧_𝑗=𝑑_∗𝑗−𝑧_𝑗,ifΔ_𝑗>0,Δ𝑧_𝑗=𝑑^∗_𝑗−𝑧_𝑗,ifΔ_𝑗𝑑<0,_∗𝑗−𝑧_𝑗≤Δ𝑧_𝑗≤𝑑^∗_𝑗−𝑧_𝑗,ifΔ_𝑗(𝑡)=0,𝑗∈𝐽Δ𝑢=𝑓_∗(𝑡)(𝑡)(𝑡)−𝑢,ifΔ>0,Δ𝑢=𝑓^∗(𝑡)(𝑡)𝑓−𝑢,ifΔ<0,_∗≤Δ𝑢(𝑡)≤𝑓^∗,ifΔ(𝑡)=0,𝑡∈𝑇, (5.2) and is equal to𝛽=𝛽𝑣,𝑆_𝐵=_{𝑗∈𝐽+𝐻}Δ_𝑗𝑧_𝑗−𝑑_∗𝑗+_{𝑗∈𝐽−𝐻}Δ_𝑗𝑧_𝑗−𝑑^∗_𝑗+_𝑡∈𝑇+Δ(𝑡)𝑢(𝑡)−𝑓_∗+_{𝑡∈𝑇−}Δ(𝑡)𝑢(𝑡)−𝑓^∗, (5.3) where𝑇⁺=𝑡∈𝑇_𝐻,Δ(𝑡)>0,𝑇⁻=𝑡∈𝑇_𝐻(𝑡),𝐽,Δ<0_+𝐻=𝑗∈𝐽_𝐻,Δ_𝑗>0,𝐽⁻_𝐻=𝑗∈𝐽_𝐻,Δ_𝑗<0. (5.4) The number 𝛽(𝑣,𝑆_𝐵) is called a value of suboptimality of the support control {𝑣,𝑆_𝐵}. From this, 𝐽(𝑣)−𝐽(𝑣)≤𝛽(𝑣,𝑆_𝐵). From this inequality, we deduce the following results.

6. Optimality and 𝜺-Optimality Criterion [8–10]

Theorem 6.1. The following relations: 𝑢(𝑡)=𝑓_∗(𝑡)𝑢(𝑡),ifΔ>0,=𝑓^∗(𝑡)𝑓,ifΔ<0,_∗≤𝑢(𝑡)≤𝑓^∗(𝑡)𝑧,ifΔ=0,𝑡∈𝑇,_𝑗=𝑑_∗𝑗,ifΔ_𝑗𝑧>0,_𝑗=𝑑^∗_𝑗,ifΔ_𝑗𝑑<0,_∗𝑗≤𝑧_𝑗≤𝑑^∗_𝑗,ifΔ_𝑗=0,𝑗∈𝐽, (6.1) are sufficient, and in the cases of nondegeneracy, they are necessary for the optimality of support control {𝑣,𝑆_𝐵}.

Theorem 6.2. For any 𝜀≥0, the admissible control 𝑣 is 𝜀-optimal if and only if there exists a support 𝑆_𝐵 such that 𝛽(𝑣,𝑆_𝐵)≤𝜀.

7. Numerical Algorithm for Solving the Problem

Let it be said that 𝜀>0 is a given number. Suppose that criterion optimality and 𝜀-optimality do not satisfy an initial support control {𝑣,𝑆_𝐵}. From this we let it pass to iteration of the algorithm: {𝑣,𝑆_𝐵} for the “new” {𝑣,𝑆_𝐵} so that 𝛽(𝑣,𝑆_𝐵)≤𝛽(𝑣,𝑆_𝐵). The iteration consists in three procedures:(1)change of an admissible control 𝑣→𝑣,(2)change of support 𝑆_𝐵→𝑆_𝐵,(3)final procedure.

7.1. Change of Control

Consider an initial support control {𝑣,𝑆_𝐵}, and let 𝑣=(𝑧,𝑢) be a new admissible control constructed by the formulas: 𝑧_𝑗=𝑧_𝑗+𝜃⁰𝑙_𝑗,𝑗∈𝐽,𝑢(𝑡)=𝑢(𝑡)+𝜃⁰𝑙(𝑡),𝑡∈𝑇_ℎ, (7.1) where 𝑙=(𝑙_𝑗,𝑗∈𝐽,𝑙(𝑡),𝑡∈𝑇_ℎ) is an admissible direction of changing a control 𝑣; 𝜃⁰ is the maximum step along this direction.

Construct the Admissible Direction
Let us introduce a pseudocontrol 𝑣=(𝑧,𝑢(𝑡),𝑡∈𝑇).
First, we compute the nonsupport values of a pseudo-control 𝑧_𝑗=𝑑_𝑗∗,ifΔ_𝑗𝑑≥0,_∗𝑗,ifΔ_𝑗≤0,𝑗∈𝐽_𝐻(𝑡)=𝑓;𝑢^∗(𝑡)𝑓,ifΔ≤0,_∗,ifΔ(𝑡)≥0,𝑡∈𝑇_𝐻. (7.2)
Secondly, support values of a pseudocontrol {𝑧_𝑗,𝑗∈𝐽_𝐵;𝑢(𝑡),𝑡∈𝑇_𝐵} are computed from the equations: _{𝑗∈𝐽𝐵}𝐷𝐼,𝑗𝑧_𝑗+_{𝑡∈𝑇𝐵}𝑑(𝑡)𝑢(𝑡)=𝑔−_{𝑗∈𝐽𝐻}𝐷𝐼,𝑗𝑧_𝑗+_{𝑡∈𝑇𝐻}𝑑(𝑡)𝑢_{𝑗∈𝐽𝐵}(𝑡),𝐺𝐿,𝑗𝑧_𝑗=𝛾−_{𝑗∈𝐽𝐻}𝐺𝐿,𝑗𝑧_𝑗. (7.3) By a pseudocontrol we compute the admissible direction 𝑙∶𝑙_𝑗=𝑧_𝑗−𝑧_𝑗,𝑗∈𝐽;𝑙(𝑡)=𝑢(𝑡)−𝑢(𝑡),𝑡∈𝑇_ℎ. (7.4)

Construct the Maximal Step
Since 𝑣 is to be admissible, then we have 𝑑_∗≤𝑧≤𝑑^∗;𝑓_∗≤𝑢(𝑡)≤𝑓^∗,𝑡∈𝑇_ℎ, (7.5) that is, 𝑑_∗≤𝑧_𝑗+𝜃⁰𝑙_𝑗≤𝑑^∗𝑓,𝑗∈𝐽;_∗≤𝑢(𝑡)+𝜃⁰𝑙(𝑡)≤𝑓^∗,𝑡∈𝑇_ℎ. (7.6) Then, the maximal step 𝜃⁰ is chosen as 𝜃⁰=min{1;𝜃(𝑡₀);𝜃_𝑗0}. Here 𝜃_𝑗0=min𝜃_𝑗: 𝜃_𝑗=𝑑^∗_𝑗−𝑧_𝑗𝑙_𝑗,if𝑙_j𝑑>0,_∗𝑗−𝑧_𝑗𝑙_𝑗,if𝑙_𝑗<0,+∞,if𝑙_𝑗=0,𝑗∈𝐽_𝐵. (7.7)𝜃(𝑡₀)=min_{𝑡∈𝑇𝐵}𝜃(𝑡): 𝜃(𝑡)=𝑓^∗−𝑢(𝑡)𝑙(𝑡)(𝑡)𝑓,if𝑙>0,_∗−𝑢(𝑡)𝑙(𝑡)(𝑡)(𝑡),if𝑙<0,+∞,if𝑙=0,𝑡∈𝑇_𝐵. (7.8) Let us calculate the value of suboptimality of the new support control {𝑣,𝑆_𝐵}, with 𝑣 computed according to (7.1): 𝛽(𝑣,𝑆_𝐵)=(1−𝜃⁰)𝛽(𝑣,𝑆_𝐵).
Consequentlyif 𝜃⁰=1, then 𝑣 is an optimal control;if 𝛽(𝑣,𝑆_𝐵)≤𝜀, then 𝑣 is an 𝜀-optimal control;if 𝛽(𝑣,𝑆_𝐵)>𝜀, then we perform a change of support.

7.2. Change of Support

The change of support 𝑆_𝐵→𝑆_𝐵 will be to satisfy 𝛽(𝑣,𝑆_𝐵)<𝛽(𝑣,𝑆_𝐵).

Here, we have 𝜃⁰=min(𝜃(𝑡₀),𝑡₀∈T_𝐵;𝜃_𝑗0,𝑗₀∈𝐽_𝐵).

We will distinguish between two cases which can occur after the first procedure:(a)𝜃⁰=𝜃_𝑗0, 𝑗₀∈𝐽_𝐵.(b)𝜃⁰=𝜃(𝑡₀), 𝑡₀∈𝑇_𝐵.

Each case is investigated separately.

This change is based on variation of potentials, estimates, and cocontrol:𝜈^′=𝜈+Δ𝜈;Δ_𝑗=Δ_𝑗+𝜎⁰𝛿_𝑗,𝑗∈𝐽;Δ(𝑡)=Δ(𝑡)+𝜎⁰𝛿(𝑡),𝑡∈𝑇_ℎ, (7.9) where (𝛿_𝑗,𝑗∈𝐽,𝛿(𝑡),𝑡∈𝑇_ℎ) is an admissible direction of change (Δ,Δ(⋅)), 𝜎⁰ a maximal step along this direction, and Δ𝑣 increment of potential.

Construct an Admissible Direction (𝛿_𝑗,𝑗∈𝐽,𝛿(𝑡),𝑡∈𝑇_ℎ)
First, Construct the support values 𝛿_𝐵=(𝛿_𝑗,𝑗∈𝐽_𝐵,𝛿(𝑡),𝑡∈𝑇_𝐵) of admissible direction for each case.

Case a (𝜃⁰=𝜃_𝑗0). Let us put 𝛿(𝑡)=0,if𝑡∈𝑇_𝐵,𝛿_𝑗=0,if𝑗≠𝑗₀,𝑗∈𝐽_𝐵,𝛿_𝑗0=1,if𝑧_𝑗0=𝑑_∗𝑗0,𝛿_𝑗0=−1,if𝑧_𝑗0=𝑑^∗_𝑗0. (7.10)

Case b (𝜃⁰=𝜃(𝑡₀)). Let us put 𝛿_𝑗=0,if𝑗∈𝐽_𝐵,𝛿(𝑡)𝑇=0,if𝑡∈^𝐵𝑡₀,𝛿𝑡₀=1,if𝑢𝑡₀=𝑓_∗,𝛿𝑡₀=−1,if𝑢𝑡₀=𝑓^∗. (7.11) By using the values 𝛿_𝐵, we compute the variation Δ𝜈=^Δ𝜈𝑢_Δ𝜈𝑧 of potentials as Δ𝜈^′=𝛿^′_𝐵𝑄.
Finally, we get the variation of nonsupport components of the estimates and the cocontrol: 𝛿_𝑗,𝑗∈𝐽_𝐻𝐷𝐺,𝛿(𝑡)=Δ𝜈′𝐼,𝑗𝐿,𝑗,𝑡∈𝑇_𝐻=Δ𝑣^′_𝑢𝑑(𝑡),𝑡∈𝑇_𝐻. (7.12)

Construct a Maximal Step 𝜎⁰
A maximal step is equal to 𝜎⁰=min(𝜎⁰_𝑗,𝜎⁰_𝑡), where 𝜎⁰_𝑗=𝜎_𝑗1=min𝜎_𝑗,𝑗∈𝐽_𝐻;𝜎⁰_𝑡=𝜎𝑡₁=min𝜎(𝑡),𝑡∈𝑇_𝐻, (7.13) where 𝜎_𝑗=−Δ_𝑗𝛿_𝑗ifΔ_𝑗𝛿_𝑗<0,+∞ifΔ_𝑗𝛿_𝑗≥0,𝑗∈𝐽_𝐻,𝜎(𝑡)=−Δ(𝑡)𝛿(𝑡)(𝑡)𝛿(𝑡)(𝑡)𝛿(𝑡)ifΔ<0,+∞ifΔ≥0,𝑡∈𝑇_𝐻. (7.14)

Construct a New Support
For constructing a new support, we consider the following cases.
(1) 𝜃⁰=𝜃(𝑡₀),𝜎⁰=𝜎(𝑡₁).
A new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵}, where 𝑇_𝐵=𝑇_𝐵𝑡₀∪𝑡₁,𝐽_𝐵=𝐽_𝐵. (7.15)
(2) 𝜃⁰=𝜃(𝑡₀),𝜎⁰=𝜎_𝑗1.
A new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵}, where 𝑇_𝐵=𝑇_𝐵𝑡₀,𝐽_𝐵=𝐽_𝐵∪𝑗₁. (7.16)
(3) 𝜃⁰=𝜃_𝑗0,𝜎⁰=𝜎_𝑗1.
A new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵}, where 𝑇_𝐵=𝑇_𝐵,𝐽_𝐵=𝐽_𝐵𝑗₀∪𝑗₁. (7.17)
(4) 𝜃⁰=𝜃_𝑗0,𝜎⁰=𝜎(𝑡₁).
A new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵}, where 𝑇_𝐵=𝑇_𝐵∪𝑡₁,𝐽_𝐵=𝐽_𝐵𝑗₀. (7.18)
A value of suboptimality for support control 𝛽(𝑣,𝑆_𝐵) is equal to 𝛽𝑣,𝑆_𝐵=1−𝜃⁰𝛽𝑣,𝑆_𝐵−𝛼𝜎⁰, (7.19) where 𝑧𝛼=^𝑗0−𝑧_𝑗0,if𝜃⁰=𝜃_𝑗0,𝑢𝑡₀−𝑢𝑡₀,if𝜃⁰=𝜃𝑡₀. (7.20)(1)If 𝛽(𝑣,𝑆_𝐵)=0, then the control 𝑣 is optimal for problem (2.1)–(2.4).(2)If 𝛽(𝑣,𝑆_𝐵)<𝜀, then the control 𝑣 is 𝜀-optimal for problem (2.1)–(2.4).(3)If 𝛽(𝑣,𝑆_𝐵)>𝜀, then we pass to a new iteration with the support control {𝑣,𝑆_𝐵} or to the final procedure.

7.3. Final Procedure

By using a support 𝑆_𝐵, we construct a quasicontrol 𝑣=(𝑧,𝑢(𝑡),𝑡∈𝑇):𝑧_𝑗=𝑑_𝑗∗,ifΔ_𝑗𝑑>0,_∗𝑗,ifΔ_𝑗∈𝑑<0,_𝑗∗,𝑑^∗_𝑗,ifΔ_𝑗=0,𝑗∈𝐽_𝐻(𝑡)=𝑓;𝑢^∗(𝑡)𝑓,ifΔ<0,_∗,ifΔ(𝑡)>0,𝑡∈𝑇_𝐻. (7.21) If𝐷(𝐼,𝐽)𝑧+^𝑡∗₀𝜑(𝑡)𝑢(𝑡)𝑑𝑡=𝑔,𝐺(𝐿,𝐽)𝑧=𝛾, (7.22) then 𝑣 is optimal control, and if𝐷(𝐼,𝐽)𝑧+^𝑡∗₀𝜑(𝑡)𝑢(𝑡)𝑑𝑡≠𝑔,𝐺(𝐿,𝐽)𝑧≠𝛾, (7.23) then denote 𝑇⁰={𝑡_𝑖,𝑖=1,𝑠},  𝑠=|𝑇_𝐵|.

Here, 𝑡_𝑖,𝑖=1,𝑠 are zeroes of the optimal cocontrol Δ(𝑡)=0, 𝑡∈𝑇; 𝑡₀=0, 𝑡_𝑠+1=𝑡^∗. Suppose thatΔ̇𝑡_𝑖≠0,𝑖=1,𝑠. (7.24) From system (7.23), we deduce and construct the following function:𝑓(Θ)=𝐷𝐼,𝐽_𝐵𝑧𝐽_𝐵+𝐷𝐼,𝐽_𝐻𝑧𝐽_𝐻+^𝑠_𝑖=0𝑓^∗+𝑓_∗2−𝑓^∗−𝑓_∗2signΔ̇𝑡_𝑖^𝑡𝑖+1_𝑡𝑖𝜑(𝑡)𝐺𝑑𝑡−𝑔𝐿,𝐽_𝐵𝑧𝐽_𝐵+𝐺𝐿,𝐽_𝐻𝑧𝐽_𝐻,−𝛾 (7.25) where𝑧_𝑗=𝑑^∗_𝑗+𝑑_𝑗∗2−𝑑^∗_𝑗−𝑑_𝑗∗2signΔ_𝑗,𝑗∈𝐽_𝐻,Θ=𝑡_𝑖,𝑖=1,𝑠;𝑧_𝑗,𝑗∈𝐽_𝐵. (7.26) The final procedure is to find the solutionΘ⁰=𝑡⁰_𝑖,𝑖=1,𝑠;𝑧⁰_𝑗,𝑗∈𝐽_𝐵 (7.27) of the system of 𝑚+𝑙-nonlinear equations𝑓(Θ)=0. (7.28) We solve this system by the Newton method using an initial approximation:Θ⁽⁰⁾=𝑡_𝑖,𝑖=1,𝑠;𝑧_𝑗,𝑗∈𝐽_𝐵. (7.29) The (𝑘+1)th approximation Θ^(𝑘+1), in step 𝑘+1≥1, is equal toΘ^(𝑘+1)=Θ^(𝑘)+ΔΘ^(𝑘),ΔΘ^(𝑘)=−𝜕𝑓⁻¹Θ^(𝑘)𝜕Θ^(𝑘)⋅𝑓Θ^(𝑘), (7.30) where𝜕𝑓Θ^(𝑘)𝜕Θ^(𝑘)=𝐷𝐼,𝐽_𝐵𝑓_∗−𝑓^∗signΔ̇𝑡_𝑖^(𝑘)𝜑𝑡_𝑖^(𝑘),𝑖=𝐺1,𝑠𝐿,𝐽_𝐵0. (7.31) As det𝑃_𝐵≠0, we can easily show thatΘdet𝜕𝑓⁽⁰⁾𝜕Θ⁽⁰⁾≠0. (7.32) For all instants 𝑡_𝑖∈𝑇_𝐵, there exists a small 𝜇>0 that all 𝑡_𝑖̃∈[𝑡_𝑖−𝜇,𝑡_𝑖+𝜇], 𝑖=1,𝑠, the matrices (𝜑(𝑡_𝑖̃),𝑖=1,𝑠) are not degenerate, and the matrix 𝜕𝑓(Θ^(𝑘))/𝜕Θ^(𝑘) is not degenerate. If elements 𝑡_𝑖^(𝑘),  𝑖=1,𝑠,  𝑘=1,2,…, do not leave the 𝜇-vicinity of 𝑡_𝑖, 𝑖=1,𝑠, vector Θ^(𝑘∗) is taken as a solution of (7.28) provided that𝑓Θ^(𝑘∗)≤𝜂, (7.33) for a given 𝜂>0. So we put 𝜃⁰=𝜃^(𝑘∗). The suboptimal control for problem (2.1)–(2.4) is computed as𝑧⁰_𝑗=𝑧⁰_𝑗,𝑗∈𝐽_𝐵,𝑧_𝑗,𝑗∈𝐽_𝐻;𝑢⁰(𝑡)=𝑓^∗+𝑓_∗2−𝑓^∗−𝑓_∗2signΔ̇𝑡⁰_𝑖,𝑡∈𝑡⁰_𝑖,𝑡⁰_𝑖+1,𝑖=1,𝑠. (7.34) If the Newton method does not converge, we decrease parameter ℎ>0 and perform the iterative process again.

8. Example

We illustrate the results obtained in this paper using the following example:₀²⁵𝑢(𝑡)𝑑𝑡⟶min,𝑥̇₁=𝑥₃,𝑥̇₂=𝑥₄,𝑥̇₃=−𝑥₁+𝑥₂+𝑢,𝑥̇₄=0.1𝑥₁−1.01𝑥₂,𝑥₁(25)=𝑥₂(25)=𝑥₃(25)=𝑥₄()(𝑡)[].25=0,0≤𝑢≤1,𝑡∈0,25 (8.1) Let the matrices and arrays be as follows:0000,21𝐻=1000010000100001,𝑔=,𝐺=1000010000100001𝛾=0.10.25,𝑑_∗=0000,𝑑^∗=2222. (8.2) Let us consider the initial condition as𝑥₁(0)=0.1,𝑥₂(0)=0.25,𝑥₃(0)=2,𝑥₄(0)=1. (8.3)

Problem (8.1) is reduced to a canonical form (2.1)–(2.4) by introducing the new variable 𝑥̇₅=𝑢,  𝑥₅(0)=0. Then, the control criterion takes the form −𝑥₅(𝑡^∗)→max. In the class of discrete controls with quantization period ℎ=25/1000=0.025, problem (8.1) is equivalent to LP problem of dimension 4×1000.

To construct the optimal open-loop control of problem (8.1).

As an initial support, a set 𝑇_𝐵={5,10,15,20} was selected. This support corresponds to the set nonsupport zeroes of the cocontrol 𝑇_𝑛0={3.725,9.725,15.3,21.3}. The problem was solved in 18 iterations; that is, to construct the optimal open-loop control, a support 4×4-matrix was changed 18 times. The optimal value of the control criterion was found to be equal 6.602499 and the time is very quickly 2.30.

Movements of the cocontrol Δ(𝑡) in the course of iterations are pictured in Figure 1.

Figure 1 

The given data illustrate the effectiveness of the method used. In our opinion, the time it takes today to construct optimal open-loop controls is not of significant importance. It is only important that the method is able to construct a reliable solution in a reasonable time. Let us give some calculations.

At first, a characteristic of the methods for comparison is chosen. A comparison of the number of iterations in various methods is not always reasonable as iterations of various methods often differ a great deal from one another. It is more naturel to define the effectiveness of method [19] by using the number on integration of a primal or an adjoint system with insignificant volume or required operative memory. In this connection, as a unit of the complexity the time of integration of a primal or an adjoint system on the whole control interval 𝑇 is taken. If a method admits to make operation in parallel, then the complexity is defined by the time needed for a set of microprocessors to solve the problem.

The proposed characteristic is not absolute (exact) as it does not take into account to evaluate methods at “first approximation.” Table 1 contains some information on the solution to problem (8.1) for other quantization periods.

Table 1 

Of course, one can solve problem (8.1) by LP methods, transforming the problem (4.2)–(4.5). In doing so, one integration of the system is sufficient to form the matrix of the LP problem. However, such “static” approach is concerned with a large volume of required operative memory, and it is fundamentally different from the traditional “dynamical” approaches based on dynamical models (2.1)–(2.4). Then, problem (2.1)–(2.4) was solved.

In Figure 2, The realization 𝑢^∗(𝜏), 𝜏∈𝑇_ℎ, is given. In Figure 3, projections of transients of system (8.1) closed by optimal open-loop on planes 𝑥₁𝑥₃ are presented. In Figure 4, projections of transients of system (8.1) closed by optimal open-loop on planes 𝑥₂𝑥₄ are presented.

Figure 2 

Figure 3 

Figure 4 

The optimal initial state is𝑥₁(0)=0.1009729,𝑥₂(0)=0.2502507,𝑥₃(0)=0.9933905,𝑥₄(0)=1.0010008. (8.4)

9. Conclusion

An optimal control problem with free initial condition has been considered.

The model problem becomes a problem, where we search the best of initial condition and a control which permits to bring the system of initial condition 𝑥₀∈𝑋₀ towards the final state which verifies the constraint 𝐻𝑥(𝑡^∗)=𝑔.

To conclude, it appears that the study and applications of adaptive methods have at least important advantage. Control law computations can be executed very quickly in real time, in particular, by using parallel computers.