Adaptive Method for Solving Optimal Control Problem with State and Control Variables

Abstract

The problem of optimal control with state and control variables is studied. The variables are: a scalar vector 𝑥 and the control 𝑢(𝑡); these variables are bonded, that is, the right-hand side of the ordinary differential equation contains both state and control variables in a mixed form. For solution of this problem, we used adaptive method and technology of linear programming.

1. Introduction

Problems of optimal control have been intensively investigated in the world literature for over forty years. During this period, a series of fundamental results have been obtained, among which should be noted the maximum principle [1] and dynamic programming [2, 3]. Results of the theory were taken up in various fields of science, engineering, and economics.

The optimal control problem with mixed variables and free terminal time is considered. This problem is among the most difficult problems in the mathematical theory of control processes [4–7]. An algorithm based on the concept of simplex method [4, 5, 8, 9] so called support control is proposed to solve this problem.

The aim of the paper is to realize the adaptive method of linear programming [8]. In our opinion the numerical solution is impossible without using the computers of discrete controls defined on the quantized axes as accessible controls. This made, it possible to eliminate some analytical problems and reduce the optimal control problem to a linear programming problem. The obtained results show that the adequate consideration of the dynamic structure of the problem in question makes it possible to construct very fast algorithms of their solution.

The work has the following structure. In Section 2, The terminal optimal control problem with mixed variables is formulated. In Section 3, we give some definitions needed in this paper. In Section 4, the definition of support is introduced. Primal and dual ways of its dynamical identification are given. In Section 5, we calculate a value of suboptimality. In Section 6, optimality and ɛ-optimality criteria are defined. In Section 7, there is a numerical algorithm for solving the problem; the iteration consists in two procedures: change of control and change of a support to find a solution of discrete problem; at the end, we used a final procedure to find a solution in the class of piecewise continuous functions. In Section 8, the results are illustrated with a numerical example.

2. Problem Statement

We consider linear optimal control problem with control and state constraints: 𝐽(𝑥,𝑢)=𝑔𝑥𝑡_𝑓+^𝑡𝑓₀(𝐶𝑥(𝑡)+𝐷𝑢(𝑡))𝑑𝑡⟶max_𝑥,𝑢,(2.1) subject to 𝑥̇=𝑓(𝑥(𝑡)𝑥,(𝑢0()𝑡)=)=𝑥₀𝐴𝑥(𝑡)+𝐵𝑢(𝑡),0≤𝑡≤𝑡_𝑓,𝑥_min,𝑥𝑡_𝑓=𝑥_𝑓,≤𝑥(𝑡)≤𝑥_max,𝑢_min≤𝑢(𝑡)≤𝑢_max,𝑡∈𝑇=0,𝑡_𝑓,(2.2) where 𝐴,𝐵,𝐶, and 𝐷 are constant or time-dependent matrices of appropriate dimensions, 𝑥∈𝑅^𝑛 is a state of control system (2.1)–(2.2), and 𝑢(⋅)=(𝑢(𝑡),𝑡∈𝑇), 𝑇=[0,𝑡_𝑓], is a piecewise continuous function. Among these problems in which state and control are variables, we consider the following problem: 𝐽(𝑥,𝑢)=𝑐^𝑥+^𝑡𝑓₀𝑐(𝑡)𝑢(𝑡)𝑑𝑡⟶max_𝑥,𝑢,(2.3) subject to 𝐴𝑥+^𝑡∗₀ℎ(𝑥𝑡)(𝑢0()𝑡)=𝑑𝑥𝑡,₀0≤𝑡≤𝑡_𝑓,(2.4)𝑥,(2.5)_min≤𝑥(𝑡)≤𝑥_max,𝑢_min≤𝑢(𝑡)≤𝑢_max,𝑡∈𝑇=0,𝑡_𝑓,(2.6) where 𝑥∈𝑅^𝑛 is a state of control system (2.3)–(2.6); 𝑢(⋅)=(𝑢(𝑡),𝑡∈𝑇), 𝑇=[0,𝑡_𝑓], is a piecewise continuous function, 𝐴∈𝑅^𝑚×𝑛; 𝑐=𝑐(𝐽)=(𝑐_𝑗,𝑗∈𝐽); 𝑔=𝑔(𝐼)=(𝑔_𝑖,𝑖∈𝐼) is an 𝑚-vector; 𝑐(𝑡), 𝑡∈𝑇, is a continuous scalar function; ℎ(𝑡), 𝑡∈𝑇, is an 𝑚-vector function; 𝑢_min,𝑢_max are scalars; 𝑥_min=𝑥_min(𝐽)=(𝑥_min𝑗,𝑗∈𝐽), 𝑥_max=𝑥_max(𝐽)=(𝑥_max𝑗,𝑗∈𝐽) are 𝑛-vectors; 𝐼={1,…,𝑚}, 𝐽={1,…,𝑛} are sets of indices.

3. Essentials Definitions

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

Definition 3.2. The constraint (2.4) is assumed to be controllable, that is for any 𝑚-vector 𝑔, there exists a pair 𝑣, for which the equality (2.4) is fulfilled.
A generalized control 𝑣=(𝑥,𝑢(⋅)) is said to be an admissible control if it satisfies constraints (2.4)–(2.6).

Definition 3.3. An admissible control 𝑣⁰=(𝑥⁰,𝑢⁰(⋅)) is said to be an optimal open-loop control if a control criterion reaches its maximal value 𝐽𝑣⁰=max_𝑣𝐽(𝑣).(3.1)

Definition 3.4. For a given 𝜀≥0, an 𝜀-optimal control 𝑣^𝜀=(𝑥^𝜀,𝑢^𝜀(⋅)) is defined by the inequality 𝐽𝑣⁰−𝐽(𝑣^𝜀)≤𝜀.(3.2)

4. Support and the Accompanying Elements

Let us introduce a discretized time set 𝑇_ℎ={0,ℎ,…,𝑡_𝑓−ℎ} where ℎ=𝑡_𝑓/𝑁, and 𝑁 is an integer. A function 𝑢(𝑡), 𝑡∈𝑇, is called a discrete control if 𝑢(𝑡)=𝑢(𝜏),𝑡∈[𝜏,𝜏+ℎ),𝜏∈𝑇_ℎ.(4.1) First, we describe a method of computing the solution of problem (2.3)–(2.6) in the class of discrete control, and then we present the final procedure which uses this solution as an initial approximation for solving problem (2.3)–(2.6) in the class of piecewise continuous functions.

Definitions of admissible, optimal, 𝜀-optimal controls for discrete functions are given in a standard form.

Choose an arbitrary subset 𝑇_𝐵⊂𝑇_ℎ of 𝑙≤𝑚 elements and an arbitrary subset 𝐽_𝐵⊂𝐽 of 𝑚-𝑙 elements.

Form the matrix, 𝑃_𝐵=𝑎_𝑗=𝐴(𝐼,𝑗),𝑗∈𝐽_𝐵;𝑑(𝑡),𝑡∈𝑇_𝐵,(4.2) where 𝑑(𝑡)=∫^𝑡𝑡+ℎℎ(𝑠)𝑑𝑠, 𝑡∈𝑇_ℎ.

A set 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} is said to be a support of problem (2.3)–(2.6) if det𝑃_𝐵≠0.

A pair {𝑣,𝑆_𝐵} of an admissible control 𝑣(𝑡)=(𝑥,𝑢(𝑡),𝑡∈𝑇) and a support 𝑆_𝐵 is said to be a support control.

A support control {𝑣,𝑆_𝐵} is said to be primally nonsingular if 𝑑_∗𝑗<𝑥_𝑗<𝑑^∗𝑗,𝑗∈𝐽_𝐵;𝑓_∗<𝑢(𝑡)<𝑓^∗,𝑡∈𝑇_𝐵.

Let us consider another admissible control 𝑣=(𝑥,𝑢(⋅))=𝑣+Δ𝑣, where 𝑥=𝑥+Δ𝑥,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^Δ𝑥+^𝑡𝑓₀𝑐(𝑡)Δ𝑢(𝑡)𝑑𝑡.(4.3) Since 𝐴Δ𝑥+^𝑧0ℎ(𝑡)Δ𝑢(𝑡)𝑑𝑡=0,(4.4) then the increment of the functional equals Δ𝐽(𝑣)=𝑐^−𝜈^𝐴Δ𝑥+^𝑡𝑓₀𝑐(𝑡)−𝜈^ℎ(𝑡)Δ𝑢(𝑡)𝑑𝑡,(4.5) where 𝜈∈𝑅^𝑚 is called potentials: 𝜈^=𝑞^𝐵𝑄, 𝑞_𝐵=(𝑐_𝑟𝑗,𝑗∈𝐽_𝐵;𝑞(𝑡),𝑡∈𝑇_𝐵), 𝑄=𝑃^𝐵−1, 𝑞(𝑡)=∫^𝑡𝑡+ℎ𝑐(𝑠)𝑑𝑠, 𝑡∈𝑇_ℎ.

Introduce an 𝑛-vector of estimates Δ^=𝜈^𝐴−𝑐^ and a function of cocontrol Δ(⋅)=(Δ(𝑡)=𝜈^𝑑(𝑡)−𝑞(𝑡),𝑡∈𝑇_ℎ). With the use of these notions, the value of the cost functional increment takes the form Δ𝐽(𝑣)=Δ^Δ𝑥−_{𝑡∈𝑇ℎ}Δ(𝑡)Δ𝑢(𝑡).(4.6)

A support control {𝑣,𝑆_𝐵} is dually nonsingular 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: 𝑥_min−𝑥≤Δ𝑥≤𝑥_max−𝑥,𝑢_min−𝑢(𝑡)≤Δ𝑢(𝑡)≤𝑢_max−𝑢(𝑡),𝑡∈𝑇.(5.1) The maximum of functional (4.6) under constraints (5.1) is reached for: ΔΔ𝑥𝑥_𝑗𝑗=𝑥_min𝑗−𝑥_𝑗ifΔ_𝑗>0,𝑥=𝑥_max𝑗−𝑥_𝑗ifΔ_𝑗<0,_min𝑗−𝑥Δ_𝑗𝑢≤(𝑡)Δ=𝑥_𝑗𝑢≤_min𝑥_max𝑗−𝑥_𝑗ifΔ_𝑗=0,𝑗∈𝐽,Δ𝑢(𝑡)=𝑢_max−𝑢(𝑡)ifΔ(𝑡)>0𝑢−𝑢(𝑡)ifΔ(𝑡)<0_min≤Δ𝑢(𝑡)≤𝑢_maxifΔ(𝑡)=0,𝑡∈𝑇_ℎ,(5.2) and is equal to 𝛽==𝛽_{𝑗∈𝐽+𝐻}𝑣,𝑆_𝐵Δ_𝑗𝑥^𝑗−𝑥_min𝑗+_{𝑗∈𝐽−𝐻}Δ_𝑗𝑥^𝑗−𝑥_max𝑗+_𝑡∈𝑇+Δ(𝑡)𝑢(𝑡)−𝑢_min+_{𝑡∈𝑇−}Δ(𝑡)𝑢(𝑡)−𝑢_max,(5.3) where 𝑇𝐽⁺=𝑡∈𝑇_𝐻,Δ(𝑡)>0,𝑇⁻=𝑡∈𝑇_𝐻,Δ(𝑡)<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 there, 𝐽(𝑣)−𝐽(𝑣)≤𝛽(𝑣,𝑆_𝐵). Of this last inequality, the following result is deduced.

6. Optimality and 𝜀-Optimality Criterion

Theorem 6.1 (see [8]). The following relations: 𝑢𝑢((𝑡𝑡))==𝑢𝑢_minmax𝑖𝑓Δ(𝑡)>0,𝑢𝑖𝑓Δ(𝑡)<0,_min≤𝑢(𝑥𝑡)_𝑗≤𝑢_max𝑖𝑓Δ(𝑡)=0,𝑡∈𝑇_ℎ,𝑥=𝑥_min𝑗𝑖𝑓Δ_𝑗>0,_𝑗𝑥=𝑥_max𝑗𝑖𝑓Δ_𝑗<0,_min𝑗≤𝑥_𝑗≤𝑥_max𝑗𝑖𝑓Δ_𝑗=0,𝑗∈𝐽,(6.1) are sufficient, and in the case of non degeneracy, they are necessary for the optimality of control 𝑣.

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. Primal Method for Constructing the Optimal Controls

A support is used not only to identify the optimal and 𝜀-optimal controls, but also it is the main tool of the method. The method suggested is iterative, and its aim is to construct an 𝜀-solution of problem (2.3)–(2.6) for a given 𝜀≥0. As a support will be changing during the iterations together with an admissible control, it is natural to consider them as a pair.

Below to simplify the calculations, we assume that on the iterations, only primally and dually nonsingular support controls are used.

The iteration of the method is a change of an “old” control {𝑣,𝑆_𝐵} for the “new” one {𝑣,𝑆_𝐵} so that 𝛽{𝑣,𝑆_𝐵}≤𝛽{𝑣,𝑆_𝐵}. The iteration consists of two procedures: (1)change of an admissible control 𝑣→𝑣,(2)change of support 𝑆_𝐵→𝑆_𝐵. Construction of the initial support control concerns with the first phase of the method and can be performed with the use of the algorithm described below.

At the beginning of each iteration the following information is stored: (1)an admissible control 𝑣,(2)a support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵},(3)a value of suboptimality 𝛽=𝛽(𝑣,𝑆_𝐵). Before the beginning of the iteration, we make sure that a support control {𝑣,𝑆_𝐵} does not satisfy the criterion of 𝜀-optimality.

7.1. Change of an Admissible Control

The new admissible control is constructed according to the formulas: 𝑥_𝑗=𝑥_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.

7.1.1. Construct of the Admissible Direction

Let us introduce a pseudocontrol 𝑣̃=(𝑥̃,𝑢(̃𝑡),𝑡∈𝑇).

First, we compute the nonsupport values of a pseudocontrol 𝑥̃_𝑗=𝑥^min𝑗𝑥ifΔ_𝑗≥0,_max𝑗𝑢̃(𝑡)=𝑢_maxifΔ_𝑗≤0,𝑗∈𝐽_𝐻,𝑢ifΔ(𝑡)≤0,_minifΔ(𝑡)≥0,𝑡∈𝑇_𝐻.(7.2) Support values of a pseudocontrol {𝑥̃_𝑗,𝑗∈𝐽_𝐵;𝑢(̃𝑡),𝑡∈𝑇_𝐵} are computed from the equation _𝑗_∈𝐽𝐵𝐴(𝐼,𝑗)𝑥̃_𝑗+_{𝑡∈𝑇𝐵}𝑑(𝑡)𝑢̃(𝑡)=𝑔−_{𝑗∈𝐽𝐻}𝐴(𝐼,𝑗)𝑥̃_𝑗+_{𝑡∈𝑇𝐻}𝑑(𝑡)𝑢̃(𝑡).(7.3)

With the use of a pseudocontrol, we compute the admissible direction 𝑙: 𝑙_𝑗=𝑥̃_𝑗−𝑥_𝑗, 𝑗∈𝐽; 𝑙(𝑡)=𝑢(̃𝑡)−𝑢(𝑡), 𝑡∈𝑇_ℎ.

7.1.2. Construct of Maximal Step

Since 𝑣 is to be admissible, the following inequalities are to be satisfied: 𝑥_min≤𝑥≤𝑥_max;𝑢_min≤𝑢(𝑡)≤𝑢_max,𝑡∈𝑇_ℎ,(7.4) that is, 𝑢𝑥_min≤𝑥_𝑗+𝜃⁰𝑙_𝑗≤𝑥_max,𝑗∈𝐽,_min≤𝑢(𝑡)+𝜃⁰𝑙(𝑡)≤𝑢_max,𝑡∈𝑇_ℎ.(7.5) Thus, the maximal step 𝜃⁰ is chosen as 𝜃⁰=min{1;𝜃(𝑡₀);𝜃_𝑗0}.

Here, 𝜃_𝑗0=min𝜃_𝑗: 𝜃_𝑗=⎧⎪⎪⎨⎪⎪⎩𝑥^max𝑗−𝑥_𝑗𝑙_𝑗𝑥if𝑙_𝑗>0,_min𝑗−𝑥_𝑗𝑙_𝑗+∞iiff𝑙𝑙_𝑗𝑗<0,=0,𝑗∈𝐽_𝐵,(7.6) and 𝜃(𝑡₀)=min_{𝑡∈𝑇𝐵}𝜃(𝑡): 𝜃(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑢^max−𝑢(𝑡)𝑢𝑙(𝑡)if𝑙(𝑡)>0,_min−𝑢(𝑡)+∞𝑙(𝑡)iiff𝑙𝑙((𝑡𝑡))<=00,,𝑡∈𝑇_𝐵.(7.7) Let us calculate the value of suboptimality of the support control {𝑣,𝑆_𝐵} with 𝑣 computed according to (7.1): 𝛽(𝑣,𝑆_𝐵)=(1−𝜃⁰)𝛽(𝑣,𝑆_𝐵). Consequently,(1)if 𝜃⁰=1, then 𝑣 is an optimal control,(2)if 𝛽(𝑣,𝑆_𝐵)≤𝜀, then 𝑣 is an 𝜀-optimal control,(3)if 𝛽(𝑣,𝑆_𝐵)>𝜀, then we perform a change of support.

7.2. Change of Support

For 𝜀>0 given, we assume that 𝛽(𝑣,𝑆_𝐵)>𝜀 and 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.

We perform change of support 𝑆_𝐵→𝑆_𝐵 that leads to decreasing the value of suboptimality 𝛽(𝑣,𝑆_𝐵). The change of support 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,(7.8) where (𝛿_𝑗,𝑗∈𝐽,𝛿(𝑡),𝑡∈𝑇_ℎ) is an admissible direction of change (Δ,Δ(⋅)) and 𝜎⁰ is a maximal step along this direction.

7.2.1. Construct of 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

(a) 𝜃⁰=𝜃_𝑗0. Let us put 𝛿_𝑗𝛿(𝑡)=0if𝑡∈𝑇_𝐵,𝛿=_𝑗00if𝑗≠𝑗₀,𝑗∈𝐽_𝐵,𝛿_𝑗0=1if𝑥_𝑗0=𝑥_min𝑗0,=−1if𝑥_𝑗0=𝑥_max𝑗0,(7.9)

(b) 𝜃⁰=𝜃(𝑡₀). Let us put 𝛿𝛿(_𝑗𝑡)==00ifif𝑗𝑡∈∈𝐽𝑇_𝐵𝐵,𝑡₀𝛿𝑡₀,𝛿𝑡₀=1if𝑢𝑡₀=𝑢_min,=−1if𝑢𝑡₀=𝑢_max.(7.10) Using the values 𝛿_𝐵=(𝛿_𝑗,𝑗∈𝐽_𝐵,𝛿(𝑡),𝑡∈𝑇_𝐵), we compute the variation Δ𝜈 of potentials as Δ𝜈′=𝛿′_𝐵𝑄. Finally, we get the variation of nonsupport components of the estimates and the cocontrol: 𝛿𝛿_𝑗(=𝑡)Δ=𝜈Δ^𝐴𝜈_(𝐼,𝑗),𝑗∈𝐽_𝐻,𝑑(𝑡),𝑡∈𝑇_𝐻.(7.11)

7.2.2. Construct of a Maximal Step 𝜎⁰

A maximal step equals 𝜎⁰=min(𝜎^0𝑗,𝜎^0𝑡) with 𝜎^0𝑗=𝜎_𝑗1=min𝜎_𝑗,𝑗∈𝐽_𝐻;𝜎^0𝑡=𝜎(𝑡₁)=min𝜎(𝑡),𝑡∈𝑇_𝐻, where 𝜎_𝑗=⎧⎪⎨⎪⎩−Δ𝛿^𝑗_𝑗+∞iiffΔΔ_𝑗𝑗𝛿_𝑗<0,𝜎(𝑡)=−Δ(𝑡)𝛿_𝑗≥0,𝑗∈𝐽_𝐻,+∞𝛿(𝑡)iiffΔΔ((𝑡𝑡))𝛿𝛿((𝑡𝑡))<≥00,,𝑡∈𝑇_𝐻.(7.12)

7.2.3. Construct of a New Support

For constructing a new support, we consider the four following cases: (1)𝜃⁰=𝜃(𝑡₀),𝜎⁰=𝜎(𝑡₁): a new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} has two following components: 𝑇_𝐵=𝑇^𝐵𝑡₀∪𝑡¹,𝐽_𝐵=𝐽_𝐵,(7.13)(2)𝜃⁰=𝜃(𝑡₀),𝜎⁰=𝜎_𝑗1: a new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} has the two following components: 𝑇_𝐵=𝑇^𝐵𝑡₀,𝐽_𝐵=𝐽_𝐵∪𝑗₁,(7.14)(3)𝜃⁰=𝜃_𝑗0,𝜎⁰=𝜎_𝑗1: a new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} has two following components: 𝑇_𝐵=𝑇_𝐵,𝐽_𝐵=𝐽^𝐵𝑗₀∪𝑗¹,(7.15)(4)𝜃⁰=𝜃_𝑗0,𝜎⁰=𝜎(𝑡₁): a new support 𝑆_𝐵={𝑇_𝐵,𝐽_𝐵} has two following components: 𝑇_𝐵=𝑇_𝐵∪𝑡₁,𝐽_𝐵=𝐽^𝐵𝑗₀,(7.16)A value of suboptimality for support control 𝛽(𝑣,𝑆_𝐵) takes the form 𝛽𝑣,𝑆_𝐵=1−𝜃⁰𝛽𝑣,𝑆_𝐵−𝛼𝜎⁰,(7.17) where 𝛼=||𝑢̃𝑡⁰−||𝑥̃_𝑗0𝑢𝑡₀||if𝜃⁰=𝜃𝑡₀,−𝑥_𝑗0||if𝜃⁰=𝜃_𝑗0.(7.18)(1)If 𝛽(𝑣,𝑆_𝐵)>𝜀, then we perform the next iteration starting from the support control {𝑣,𝑆_𝐵}.(2)If 𝛽(𝑣,𝑆_𝐵)=0, then the control 𝑣 is optimal for problem (2.3)–(2.6) in the class of discrete controls.(3)If 𝛽(𝑣,𝑆_𝐵)<𝜀, then the control 𝑣 is 𝜀-optimal for problem (2.3)–(2.6) in the class of discrete controls. If we would like to get the solution of problem (2.3)–(2.6) in the class of piecewise continuous control, we pass to the final procedure when case 2 or 3 takes place.

7.3. Final Procedure

Let us assume that for the new control 𝑣, we have 𝛽(𝑣,𝑆_𝐵)>𝜀. With the use of the support 𝑆_𝐵 we construct a quasicontrol 𝑣̂=(𝑥̂,𝑢(̂𝑡),𝑡∈𝑇), 𝑥̂_𝑗=⎧⎪⎨⎪⎩𝑥^min𝑗𝑥ifΔ_𝑗>0,_max𝑗∈𝑥ifΔ_𝑗<0,_min𝑗𝑢̂(𝑡)=⎧⎪⎨⎪⎩𝑢_min,𝑥_max𝑗ifΔ^𝑗=0,𝑗∈𝐽.𝑢,ifΔ(𝑡)<0_max∈𝑢,ifΔ(𝑡)>0,_min,𝑢_max,ifΔ(𝑡)=0,𝑡∈𝑇_ℎ.(7.19) If 𝐴(𝐼,𝐽)𝑥̂+^𝑡𝑓₀ℎ(𝑡)𝑢̂(𝑡)𝑑𝑡=𝑔,(7.20) then 𝑣̂ is optimal, and if 𝐴(𝐼,𝐽)𝑥̂+^𝑡𝑓₀ℎ(𝑡)𝑢̂(𝑡)𝑑𝑡≠𝑔,(7.21) then denote 𝑇⁰={𝑡_𝑖∈𝑇,Δ(𝑡_𝑖)=0}, where 𝑡_𝑖 are zeros of the optimal cocontrol, that is, Δ(𝑡_𝑖)=0,𝑖=1,𝑠, with 𝑠≤𝑚. Suppose that Δ̇𝑡_𝑖≠0,𝑖=1,𝑠.(7.22) Let us construct the following function: 𝑓(Θ)=𝐴+𝐴𝐼,𝐽𝐼_𝐵,𝐽_𝐻𝑥𝐽_𝐵𝑥𝐽_𝐻+^𝑠_𝑖=0𝑢^max+𝑢_min2−𝑢^max−𝑢_min2signΔ̇𝑡^𝑖^𝑡𝑖+1_𝑡𝑖ℎ(𝑡)𝑑𝑡−𝑔,(7.23) where 𝑥_𝑗=𝑥^min𝑗+𝑥_max𝑗2−𝑥^max𝑗−𝑥_min𝑗2signΔ^𝑗Θ=𝑡_𝑖,𝑗∈𝐽_𝐻,𝑡₀=0,𝑡_𝑠+1=𝑡_𝑓,,𝑖=1,𝑠;𝑥_𝑗,𝑗∈𝐽_𝐵.(7.24) The final procedure consists in finding the solution Θ⁰=𝑡^0𝑖,𝑖=1,𝑠;𝑥^0𝑗,𝑗∈𝐽_𝐵(7.25) of the system of 𝑚 nonlinear equations 𝑓(Θ)=0.(7.26) We solve this system by the Newton method using as an initial approximation of the vector Θ⁽⁰⁾=𝑡_𝑖,𝑖=1,𝑠;𝑥_𝑗,𝑗∈𝐽_𝐵.(7.27) The (𝑘+1)^th approximation Θ^(𝑘+1), at a step 𝑘+1≥1, is computed as Θ^(𝑘+1)=Θ^(𝑘)+ΔΘ^(𝑘),ΔΘ^(𝑘)=−𝜕𝑓⁻¹Θ^(𝑘)𝜕Θ^(𝑘)⋅𝑓Θ^(𝑘).(7.28) Let us compute the Jacobi matrix for (7.26) 𝜕𝑓Θ^(𝑘)𝜕Θ^(𝑘)=𝐴𝐼,𝐽_𝐵;𝑢_min−𝑢_maxsignΔ̇𝑡^𝑖(𝑘)ℎ𝑡^𝑖(𝑘),𝑖=1,𝑠(7.29) As det𝑃_𝐵≠0, we can easily show that det𝜕𝑓Θ⁽⁰⁾𝜕Θ⁽⁰⁾≠0.(7.30)

For instants 𝑡∈𝑇_𝐵, there exists a small 𝜇>0 that for any 𝑡_𝑖̃∈[𝑡_𝑖−𝜇,𝑡_𝑖+𝜇],𝑖=1,𝑠, the matrix (ℎ(𝑡_𝑖̃),𝑖=1,𝑠) is nonsingular and the matrix 𝜕𝑓(Θ^(𝑘))/𝜕Θ^(𝑘) is also nonsingular 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 (4.6) if ‖‖𝑓Θ^(𝑘∗)‖‖≤𝜂,(7.31) for a given 𝜂>0. So we put 𝜃⁰=𝜃^(𝑘∗).

The suboptimal control for problem (2.3)–(2.6) is computed as 𝑥^0𝑗=𝑥𝑥̂^0𝑗_𝑗,𝑗∈𝐽_𝐵,,𝑗∈𝐽_𝐻𝑢⁰(𝑡)=𝑢^max+𝑢_min2−𝑢^max−𝑢_min2signΔ̇𝑡^0𝑖,𝑡∈𝑡^0𝑖,𝑡^0𝑖+1,𝑖=1,𝑠.(7.32) If the Newton method does not converge, we decrease the parameter ℎ>0 and perform the iterative process again.

8. Example

We illustrate the results obtained in this paper using the following example: ⁰²⁵𝑢(𝑡𝑥)̇₁𝑑𝑡⟶min,𝑥̇₂=𝑥₃,𝑥̇₃=𝑥₄,𝑥̇₄=−𝑥₁+𝑥₂+𝑢,𝑥₁=0.1𝑥₁−1.01𝑥₂,(0)=0𝑥.1,𝑥₂(0)=0.25,𝑥₃(0)=2,𝑥₄(0)=1,₁𝑥(25)=𝑥₂(25)=𝑥₃(25)=𝑥₄(25)=0,_min≤𝑥≤𝑥_max,0≤𝑢(𝑡)≤1,𝑡∈[0,25].(8.1)

Let the matrix be 𝐴=⎛⎜⎜⎜⎜⎜⎜⎝0−00.11−1001.0𝑥100100001⎞⎟⎟⎟⎟⎟⎟⎠,ℎ(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎝1000010000100001⎞⎟⎟⎟⎟⎟⎟⎠,𝑔=_min⎛⎜⎜⎜⎜⎜⎜⎝0000⎞⎟⎟⎟⎟⎟⎟⎠,=⎛⎜⎜⎜⎜⎜⎜⎝−−−−4444⎞⎟⎟⎟⎟⎟⎟⎠,𝑥_max=⎛⎜⎜⎜⎜⎜⎜⎝4444⎞⎟⎟⎟⎟⎟⎟⎠.(8.2)

We introduce the adjoint system which is defined as 𝜓𝜓₁₂=−𝜓₃+0.1𝜓₄,=𝜓𝜓₃₃−1.01𝜓₄,𝜓₄=𝜓₁,𝜓₁=𝜓₂,𝑡_𝑓=0,𝜓₂𝑡_𝑓=0,𝜓₃𝑡_𝑓=0,𝜓₄𝑡_𝑓=0.(8.3)

Problem (8.1) is reduced to canonical form (2.3)–(2.6) by introducing the new variable 𝑥̇₅=𝑢,𝑥₅(0)=0. Then, the control criterion takes the form −𝑥₅(𝑡_𝑓)→max. In the class of discrete controls with quantization period ℎ=25/1000=0.0025, 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 of nonsupport zeroes of the cocontrol 𝑇_𝑛0={2.956,5.4863,9.55148,12.205,17.6190,19.0372}. The problem was solved in 26 iterations, that is, to construct the optimal open-loop control, a support 4×4 matrix was changed 26 times. The optimal value of the control criterion was found to be equal to 6.602054 in time 2.92.

Table 1 contains some information on the solution of problem (8.1) for other quantization periods.

Table 1 

Of course, one can solve problem (8.1) by LP methods, transforming the problem (4.6)–(7.8). 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 (2.3)–(2.6). Then, problem (2.3)–(2.6) was solved.

In Figure 1, there are control 𝑢(𝑡) and switching function for minimum principle. In Figure 2, there is phaseportrait (𝑥₁,𝑥₃) for a system (8.1). In Figure 3, there are state variables 𝑥₁(𝑡),𝑥₂(𝑡) for a system (8.1). In Figure 3, state variables 𝑥₃(𝑡),𝑥₄(𝑡) for a system (8.1). In Figure 4, state variables 𝑥₁(𝑡),𝑥₂(𝑡) for a system (8.1).

Figure 1 

Optimal control 𝑢(𝑡) and switching function.

Figure 2 

Phaseportrait 𝑥1(𝑡),𝑥3(𝑡).

Figure 3 

Optimal state variables 𝑥1(𝑡),𝑥2(𝑡).

Figure 4 

Optimal state variables 𝑥3(𝑡),𝑥4(𝑡).