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𝑡(𝑥,𝑢)=𝑔𝑓+𝑡𝑓0(𝐶𝑥(𝑡)+𝐷𝑢(𝑡))𝑑𝑡⟶max𝑥,𝑢,(2.1) subject to ̇𝑥=𝑓(𝑥(𝑡),𝑢(𝑡))=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),0≤𝑡≤𝑡𝑓,𝑥(0)=𝑥0𝑡,𝑥𝑓=𝑥𝑓,𝑥min≤𝑥(𝑡)≤𝑥max,𝑢min≤𝑢(𝑡)≤𝑢max,𝑡∈𝑇=0,𝑡𝑓,(2.2) where 𝐴,𝐵,𝐶, and 𝐷 are constant or time-dependent matrices of appropriate dimensions, 𝑥∈𝑅𝑛 is a state of control system (2.1)–(2.2), and 𝑢(⋅)=(𝑢(𝑡),𝑡∈𝑇), 𝑇=[0,𝑡𝑓], is a piecewise continuous function. Among these problems in which state and control are variables, we consider the following problem: 𝐽(𝑥,𝑢)=ğ‘î…žî€œğ‘¥+𝑡𝑓0𝑐(𝑡)𝑢(𝑡)𝑑𝑡⟶max𝑥,𝑢,(2.3) subject to 𝐴𝑥+𝑡∗0ℎ(𝑡)𝑢(𝑡)𝑑𝑡,0≤𝑡≤𝑡𝑓,(2.4)𝑥(0)=𝑥0,𝑥(2.5)min≤𝑥(𝑡)≤𝑥max,𝑢min≤𝑢(𝑡)≤𝑢max,𝑡∈𝑇=0,𝑡𝑓,(2.6) where 𝑥∈𝑅𝑛 is a state of control system (2.3)–(2.6); 𝑢(⋅)=(𝑢(𝑡),𝑡∈𝑇), 𝑇=[0,𝑡𝑓], is a piecewise continuous function, 𝐴∈𝑅𝑚×𝑛; 𝑐=𝑐(𝐽)=(𝑐𝑗,𝑗∈𝐽); 𝑔=𝑔(𝐼)=(𝑔𝑖,𝑖∈𝐼) is an 𝑚-vector; 𝑐(𝑡), 𝑡∈𝑇, is a continuous scalar function; ℎ(𝑡), 𝑡∈𝑇, is an 𝑚-vector function; 𝑢min,𝑢max are scalars; 𝑥min=𝑥min(𝐽)=(𝑥min𝑗,𝑗∈𝐽), 𝑥max=𝑥max(𝐽)=(𝑥max𝑗,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=max𝑣𝐽(𝑣).(3.1)

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

4. Support and the Accompanying Elements

Let us introduce a discretized time set ğ‘‡â„Ž={0,ℎ,…,ğ‘¡ğ‘“âˆ’â„Ž} where ℎ=𝑡𝑓/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),ğœâˆˆğ‘‡â„Ž.(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 ğ‘‡ğµâŠ‚ğ‘‡â„Ž of 𝑙≤𝑚 elements and an arbitrary subset 𝐽𝐵⊂𝐽 of 𝑚-𝑙 elements.

Form the matrix, 𝑃𝐵=î€·ğ‘Žğ‘—=𝐴(𝐼,𝑗),𝑗∈𝐽𝐵;𝑑(𝑡),𝑡∈𝑇𝐵,(4.2) where ∫𝑑(𝑡)=𝑡𝑡+ℎℎ(𝑠)𝑑𝑠, ğ‘¡âˆˆğ‘‡â„Ž.

A set 𝑆𝐵={𝑇𝐵,𝐽𝐵} is said to be a support of problem (2.3)–(2.6) if det𝑃𝐵≠0.

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

A support control {𝑣,𝑆𝐵} is said to be primally nonsingular if 𝑑∗𝑗<𝑥𝑗<𝑑∗𝑗,𝑗∈𝐽𝐵;𝑓∗<𝑢(𝑡)<𝑓∗,𝑡∈𝑇𝐵.

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)=𝐽𝑣−𝐽(𝑣)=ğ‘î…žî€œÎ”ğ‘¥+𝑡𝑓0𝑐(𝑡)Δ𝑢(𝑡)𝑑𝑡.(4.3) Since 𝐴Δ𝑥+𝑧0ℎ(𝑡)Δ𝑢(𝑡)𝑑𝑡=0,(4.4) then the increment of the functional equals 𝑐Δ𝐽(𝑣)=î…žâˆ’ğœˆî…žğ´î€¸î€œÎ”ğ‘¥+𝑡𝑓0𝑐(𝑡)âˆ’ğœˆî…žî€¸â„Ž(𝑡)Δ𝑢(𝑡)𝑑𝑡,(4.5) where 𝜈∈𝑅𝑚 is called potentials: ğœˆî…ž=ğ‘žî…žğµğ‘„, ğ‘žğµ=(𝑐𝑟𝑗,𝑗∈𝐽𝐵;ğ‘ž(𝑡),𝑡∈𝑇𝐵), 𝑄=𝑃𝐵−1, âˆ«ğ‘ž(𝑡)=𝑡𝑡+â„Žğ‘(𝑠)𝑑𝑠, ğ‘¡âˆˆğ‘‡â„Ž.

Introduce an 𝑛-vector of estimates Δ=ğœˆî…žğ´âˆ’ğ‘î…ž and a function of cocontrol Δ(⋅)=(Δ(𝑡)=ğœˆî…žğ‘‘(𝑡)âˆ’ğ‘ž(𝑡),ğ‘¡âˆˆğ‘‡â„Ž). With the use of these notions, the value of the cost functional increment takes the form Δ𝐽(𝑣)=Î”î…žî“Î”ğ‘¥âˆ’ğ‘¡âˆˆğ‘‡â„ŽÎ”(𝑡)Δ𝑢(𝑡).(4.6)

A support control {𝑣,𝑆𝐵} is dually nonsingular if Δ(𝑡)≠0,𝑡∈𝑇𝐻,Δ𝑗≠0,𝑗∈𝐽𝐻, where 𝑇𝐻=ğ‘‡â„Ž/𝑇𝐵,𝐽𝐻=𝐽/𝐽𝐵.

5. Calculation of the Value of Suboptimality

The new control 𝑣(𝑡) is admissible, if it satisfies the constraints: 𝑥min−𝑥≤Δ𝑥≤𝑥max−𝑥,𝑢min−𝑢(𝑡)≤Δ𝑢(𝑡)≤𝑢max−𝑢(𝑡),𝑡∈𝑇.(5.1) The maximum of functional (4.6) under constraints (5.1) is reached for: Δ𝑥𝑗=𝑥min𝑗−𝑥𝑗ifΔ𝑗>0,Δ𝑥𝑗=𝑥max𝑗−𝑥𝑗ifΔ𝑗𝑥<0,min𝑗−𝑥𝑗≤Δ𝑥𝑗≤𝑥max𝑗−𝑥𝑗ifΔ𝑗=0,𝑗∈𝐽,Δ𝑢(𝑡)=𝑢min−𝑢(𝑡)ifΔ(𝑡)>0Δ𝑢(𝑡)=𝑢max𝑢−𝑢(𝑡)ifΔ(𝑡)<0min≤Δ𝑢(𝑡)≤𝑢maxifΔ(𝑡)=0,ğ‘¡âˆˆğ‘‡â„Ž,(5.2) and is equal to 𝛽=𝛽𝑣,𝑆𝐵=𝑗∈𝐽+𝐻Δ𝑗𝑥𝑗−𝑥min𝑗+𝑗∈𝐽−𝐻Δ𝑗𝑥𝑗−𝑥max𝑗+𝑡∈𝑇+Δ𝑢(𝑡)(𝑡)−𝑢min+𝑡∈𝑇−𝑢Δ(𝑡)(𝑡)−𝑢max,(5.3) where 𝑇+=𝑡∈𝑇𝐻,Δ(𝑡)>0,𝑇−=𝑡∈𝑇𝐻,𝐽,Δ(𝑡)<0+𝐻=𝑗∈𝐽𝐻,Δ𝑗>0,𝐽−𝐻=𝑗∈𝐽𝐻,Δ𝑗.<0(5.4)

The number 𝛽(𝑣,𝑆𝐵) is called a value of suboptimality of the support control {𝑣,𝑆𝐵}. From there, 𝐽(𝑣)−𝐽(𝑣)≤𝛽(𝑣,𝑆𝐵). Of this last inequality, the following result is deduced.

6. Optimality and 𝜀-Optimality Criterion

Theorem 6.1 (see [8]). The following relations: 𝑢(𝑡)=𝑢min𝑢𝑖𝑓Δ(𝑡)>0,(𝑡)=𝑢max𝑢𝑖𝑓Δ(𝑡)<0,min≤𝑢(𝑡)≤𝑢max𝑖𝑓Δ(𝑡)=0,ğ‘¡âˆˆğ‘‡â„Ž,𝑥𝑗=𝑥min𝑗𝑖𝑓Δ𝑗𝑥>0,𝑗=𝑥max𝑗𝑖𝑓Δ𝑗𝑥<0,min𝑗≤𝑥𝑗≤𝑥max𝑗𝑖𝑓Δ𝑗=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 𝑆𝐵 such that 𝛽(𝑣,𝑆𝐵)≤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,𝑆𝐵} for the “new” one {𝑣,𝑆𝐵} so that 𝛽{𝑣,𝑆𝐵}≤𝛽{𝑣,𝑆𝐵}. The iteration consists of two procedures: (1)change of an admissible control 𝑣→𝑣,(2)change of support 𝑆𝐵→𝑆𝐵. 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 𝑆𝐵={𝑇𝐵,𝐽𝐵},(3)a value of suboptimality 𝛽=𝛽(𝑣,𝑆𝐵). Before the beginning of the iteration, we make sure that a support control {𝑣,𝑆𝐵} 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: 𝑥𝑗=𝑥𝑗+𝜃0𝑙𝑗,𝑗∈𝐽,𝑢(𝑡)=𝑢(𝑡)+𝜃0𝑙(𝑡),ğ‘¡âˆˆğ‘‡â„Ž,(7.1) where 𝑙=(𝑙𝑗,𝑗∈𝐽,𝑙(𝑡),ğ‘¡âˆˆğ‘‡â„Ž) 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 ̃𝑣=(̃𝑥,̃𝑢(𝑡),𝑡∈𝑇).

First, we compute the nonsupport values of a pseudocontrol ̃𝑥𝑗=𝑥min𝑗ifΔ𝑗𝑥≥0,max𝑗ifΔ𝑗≤0,𝑗∈𝐽𝐻,𝑢̃𝑢(𝑡)=max𝑢ifΔ(𝑡)≤0,minifΔ(𝑡)≥0,𝑡∈𝑇𝐻.(7.2) Support values of a pseudocontrol {̃𝑥𝑗,𝑗∈𝐽𝐵;̃𝑢(𝑡),𝑡∈𝑇𝐵} are computed from the equation 𝑗∈𝐽𝐵𝐴(𝐼,𝑗)̃𝑥𝑗+𝑡∈𝑇𝐵𝑑(𝑡)̃𝑢(𝑡)=𝑔−𝑗∈𝐽𝐻𝐴(𝐼,𝑗)̃𝑥𝑗+𝑡∈𝑇𝐻𝑑(𝑡)̃𝑢(𝑡).(7.3)

With the use of a pseudocontrol, we compute the admissible direction 𝑙: 𝑙𝑗=̃𝑥𝑗−𝑥𝑗, 𝑗∈𝐽; 𝑙(𝑡)=̃𝑢(𝑡)−𝑢(𝑡), ğ‘¡âˆˆğ‘‡â„Ž.

7.1.2. Construct of Maximal Step

Since 𝑣 is to be admissible, the following inequalities are to be satisfied: 𝑥min≤𝑥≤𝑥max;𝑢min≤𝑢(𝑡)≤𝑢max,ğ‘¡âˆˆğ‘‡â„Ž,(7.4) that is, 𝑥min≤𝑥𝑗+𝜃0𝑙𝑗≤𝑥max𝑢,𝑗∈𝐽,min≤𝑢(𝑡)+𝜃0𝑙(𝑡)≤𝑢max,ğ‘¡âˆˆğ‘‡â„Ž.(7.5) Thus, the maximal step 𝜃0 is chosen as 𝜃0=min{1;𝜃(𝑡0);𝜃𝑗0}.

Here, 𝜃𝑗0=min𝜃𝑗: 𝜃𝑗=âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘¥max𝑗−𝑥𝑗𝑙𝑗if𝑙𝑗𝑥>0,min𝑗−𝑥𝑗𝑙𝑗if𝑙𝑗<0,+∞if𝑙𝑗=0,𝑗∈𝐽𝐵,(7.6) and 𝜃(𝑡0)=min𝑡∈𝑇𝐵𝜃(𝑡): âŽ§âŽªâŽªâŽ¨âŽªâŽªâŽ©ğ‘¢ğœƒ(𝑡)=max−𝑢(𝑡)𝑢𝑙(𝑡)if𝑙(𝑡)>0,min−𝑢(𝑡)𝑙(𝑡)if𝑙(𝑡)<0,+∞if𝑙(𝑡)=0,𝑡∈𝑇𝐵.(7.7) Let us calculate the value of suboptimality of the support control {𝑣,𝑆𝐵} with 𝑣 computed according to (7.1): 𝛽(𝑣,𝑆𝐵)=(1−𝜃0)𝛽(𝑣,𝑆𝐵). Consequently,(1)if 𝜃0=1, then 𝑣 is an optimal control,(2)if 𝛽(𝑣,𝑆𝐵)≤𝜀, then 𝑣 is an 𝜀-optimal control,(3)if 𝛽(𝑣,𝑆𝐵)>𝜀, then we perform a change of support.

7.2. Change of Support

For 𝜀>0 given, we assume that 𝛽(𝑣,𝑆𝐵)>𝜀 and 𝜃0=min(𝜃(𝑡0),𝑡0∈𝑇𝐵;𝜃𝑗0,𝑗0∈𝐽𝐵). We will distinguish between two cases which can occur after the first procedure:(a)𝜃0=𝜃𝑗0,𝑗0∈𝐽𝐵,(b)𝜃0=𝜃(𝑡0),𝑡0∈𝑇𝐵.Each case is investigated separately.

We perform change of support 𝑆𝐵→𝑆𝐵 that leads to decreasing the value of suboptimality 𝛽(𝑣,𝑆𝐵). The change of support is based on variation of potentials, estimates, and cocontrol: ğœˆî…ž=𝜈+Δ𝜈;Δ𝑗=Δ𝑗+ğœŽ0𝛿𝑗,𝑗∈𝐽,Δ(𝑡)=Δ(𝑡)+ğœŽ0𝛿(𝑡),ğ‘¡âˆˆğ‘‡â„Ž,(7.8) where (𝛿𝑗,𝑗∈𝐽,𝛿(𝑡),ğ‘¡âˆˆğ‘‡â„Ž) is an admissible direction of change (Δ,Δ(⋅)) and ğœŽ0 is a maximal step along this direction.

7.2.1. Construct of an Admissible Direction (𝛿𝑗,𝑗∈𝐽,𝛿(𝑡),ğ‘¡âˆˆğ‘‡â„Ž)

First, construct the support values 𝛿𝐵=(𝛿𝑗,𝑗∈𝐽𝐵,𝛿(𝑡),𝑡∈𝑇𝐵) of admissible direction

(a) 𝜃0=𝜃𝑗0. Let us put 𝛿(𝑡)=0if𝑡∈𝑇𝐵,𝛿𝑗=0if𝑗≠𝑗0,𝑗∈𝐽𝐵,𝛿𝑗0=1if𝑥𝑗0=𝑥min𝑗0,𝛿𝑗0=−1if𝑥𝑗0=𝑥max𝑗0,(7.9)

(b) 𝜃0=𝜃(𝑡0). Let us put 𝛿𝑗=0if𝑗∈𝐽𝐵,𝑇𝛿(𝑡)=0if𝑡∈𝐵𝑡0,𝛿𝑡0=1if𝑢𝑡0=𝑢min,𝛿𝑡0=−1if𝑢𝑡0=𝑢max.(7.10) Using the values 𝛿𝐵=(𝛿𝑗,𝑗∈𝐽𝐵,𝛿(𝑡),𝑡∈𝑇𝐵), 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: 𝛿𝑗=Î”ğœˆî…žğ´(𝐼,𝑗),𝑗∈𝐽𝐻,𝛿(𝑡)=Î”ğœˆî…žğ‘‘(𝑡),𝑡∈𝑇𝐻.(7.11)

7.2.2. Construct of a Maximal Step ğœŽ0

A maximal step equals ğœŽ0=min(ğœŽ0𝑗,ğœŽ0𝑡) with ğœŽ0𝑗=ğœŽğ‘—1=minğœŽğ‘—,𝑗∈𝐽𝐻;ğœŽ0𝑡=ğœŽ(𝑡1)=minğœŽ(𝑡),𝑡∈𝑇𝐻, where ğœŽğ‘—=âŽ§âŽªâŽ¨âŽªâŽ©âˆ’Î”ğ‘—ğ›¿ğ‘—ifΔ𝑗𝛿𝑗<0,+∞ifΔ𝑗𝛿𝑗≥0,𝑗∈𝐽𝐻,îƒ¯âˆ’ğœŽ(𝑡)=Δ(𝑡)𝛿(𝑡)ifΔ(𝑡)𝛿(𝑡)<0,+∞ifΔ(𝑡)𝛿(𝑡)≥0,𝑡∈𝑇𝐻.(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),ğœŽ0=ğœŽ(𝑡1): a new support 𝑆𝐵={𝑇𝐵,𝐽𝐵} has two following components: 𝑇𝐵=𝑇𝐵𝑡0∪𝑡1,𝐽𝐵=𝐽𝐵,(7.13)(2)𝜃0=𝜃(𝑡0),ğœŽ0=ğœŽğ‘—1: a new support 𝑆𝐵={𝑇𝐵,𝐽𝐵} has the two following components: 𝑇𝐵=𝑇𝐵𝑡0,𝐽𝐵=𝐽𝐵∪𝑗1,(7.14)(3)𝜃0=𝜃𝑗0,ğœŽ0=ğœŽğ‘—1: a new support 𝑆𝐵={𝑇𝐵,𝐽𝐵} has two following components: 𝑇𝐵=𝑇𝐵,𝐽𝐵=𝐽𝐵𝑗0∪𝑗1,(7.15)(4)𝜃0=𝜃𝑗0,ğœŽ0=ğœŽ(𝑡1): a new support 𝑆𝐵={𝑇𝐵,𝐽𝐵} has two following components: 𝑇𝐵=𝑇𝐵∪𝑡1,𝐽𝐵=𝐽𝐵𝑗0,(7.16)A value of suboptimality for support control 𝛽(𝑣,𝑆𝐵) takes the form 𝛽𝑣,𝑆𝐵=1−𝜃0𝛽𝑣,ğ‘†ğµî€¸âˆ’ğ›¼ğœŽ0,(7.17) where ||𝑡𝛼=̃𝑢0−𝑢𝑡0||if𝜃0𝑡=𝜃0,||̃𝑥𝑗0−𝑥𝑗0||if𝜃0=𝜃𝑗0.(7.18)(1)If 𝛽(𝑣,𝑆𝐵)>𝜀, then we perform the next iteration starting from the support control {𝑣,𝑆𝐵}.(2)If 𝛽(𝑣,𝑆𝐵)=0, then the control 𝑣 is optimal for problem (2.3)–(2.6) in the class of discrete controls.(3)If 𝛽(𝑣,𝑆𝐵)<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,𝑆𝐵)>𝜀. With the use of the support 𝑆𝐵 we construct a quasicontrol ̂𝑣=(̂𝑥,̂𝑢(𝑡),𝑡∈𝑇), ̂𝑥𝑗=âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¥min𝑗ifΔ𝑗𝑥>0,max𝑗ifΔ𝑗∈𝑥<0,min𝑗,𝑥max𝑗ifÎ”ğ‘—âŽ§âŽªâŽ¨âŽªâŽ©ğ‘¢=0,𝑗∈𝐽.̂𝑢(𝑡)=min𝑢,ifΔ(𝑡)<0max∈𝑢,ifΔ(𝑡)>0,min,𝑢max,ifΔ(𝑡)=0,ğ‘¡âˆˆğ‘‡â„Ž.(7.19) If 𝐴(𝐼,𝐽)̂𝑥+𝑡𝑓0ℎ(𝑡)̂𝑢(𝑡)𝑑𝑡=𝑔,(7.20) then ̂𝑣 is optimal, and if 𝐴(𝐼,𝐽)̂𝑥+𝑡𝑓0ℎ(𝑡)̂𝑢(𝑡)𝑑𝑡≠𝑔,(7.21) then denote 𝑇0={𝑡𝑖∈𝑇,Δ(𝑡𝑖)=0}, where 𝑡𝑖 are zeros of the optimal cocontrol, that is, Δ(𝑡𝑖)=0,𝑖=1,𝑠, with 𝑠≤𝑚. Suppose that ̇Δ𝑡𝑖≠0,𝑖=1,𝑠.(7.22) Let us construct the following function: 𝑓(Θ)=𝐴𝐼,𝐽𝐵𝑥𝐽𝐵+𝐴𝐼,𝐽𝐻𝑥𝐽𝐻+𝑠𝑖=0𝑢max+𝑢min2−𝑢max−𝑢min2̇Δ𝑡sign𝑖𝑡𝑖+1ğ‘¡ğ‘–â„Ž(𝑡)𝑑𝑡−𝑔,(7.23) where 𝑥𝑗=𝑥min𝑗+𝑥max𝑗2−𝑥max𝑗−𝑥min𝑗2signΔ𝑗,𝑗∈𝐽𝐻,𝑡0=0,𝑡𝑠+1=𝑡𝑓,𝑡Θ=𝑖,𝑖=1,𝑠;𝑥𝑗,𝑗∈𝐽𝐵.(7.24) The final procedure consists in finding the solution Θ0=𝑡0𝑖,𝑖=1,𝑠;𝑥0𝑗,𝑗∈𝐽𝐵(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)=𝑡𝑖,𝑖=1,𝑠;𝑥𝑗,𝑗∈𝐽𝐵.(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)Θ⋅𝑓(𝑘).(7.28) Let us compute the Jacobi matrix for (7.26) Θ𝜕𝑓(𝑘)𝜕Θ(𝑘)=𝐴𝐼,𝐽𝐵;𝑢min−𝑢max̇Δ𝑡sign𝑖(𝑘)î‚â„Žî‚€ğ‘¡ğ‘–(𝑘),𝑖=1,𝑠(7.29) As det𝑃𝐵≠0, we can easily show that Θdet𝜕𝑓(0)𝜕Θ(0)≠0.(7.30)

For instants 𝑡∈𝑇𝐵, there exists a small 𝜇>0 that for any ̃𝑡𝑖∈[𝑡𝑖−𝜇,𝑡𝑖+𝜇],𝑖=1,𝑠, the matrix ̃𝑡(ℎ(𝑖),𝑖=1,𝑠) is nonsingular and the matrix 𝜕𝑓(Θ(𝑘))/𝜕Θ(𝑘) is also nonsingular if elements 𝑡𝑖(𝑘),𝑖=1,𝑠,𝑘=1,2,… do not leave the 𝜇-vicinity of 𝑡𝑖, 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 𝑥0𝑗=𝑥0𝑗,𝑗∈𝐽𝐵,̂𝑥𝑗,𝑗∈𝐽𝐻𝑢0𝑢(𝑡)=max+𝑢min2−𝑢max−𝑢min2̇Δ𝑡sign0𝑖𝑡,𝑡∈0𝑖,𝑡0𝑖+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(𝑡)𝑑𝑡⟶min,̇𝑥1=𝑥3,̇𝑥2=𝑥4,̇𝑥3=−𝑥1+𝑥2+𝑢,̇𝑥4=0.1𝑥1−1.01𝑥2,𝑥1(0)=0.1,𝑥2(0)=0.25,𝑥3(0)=2,𝑥4(𝑥0)=1,1(25)=𝑥2(25)=𝑥3(25)=𝑥4𝑥(25)=0,min≤𝑥≤𝑥max[].,0≤𝑢(𝑡)≤1,𝑡∈0,25(8.1)

Let the matrix be ⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝0000⎞⎟⎟⎟⎟⎟⎟⎠,𝑥𝐴=00100001−11000.1−1.0100,ℎ(𝑡)=1000010000100001,𝑔=min=⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠−4−4−4−4,𝑥max=⎛⎜⎜⎜⎜⎜⎜⎝4444⎞⎟⎟⎟⎟⎟⎟⎠.(8.2)

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

Problem (8.1) is reduced to canonical form (2.3)–(2.6) by introducing the new variable ̇𝑥5=𝑢,𝑥5(0)=0. Then, the control criterion takes the form −𝑥5(𝑡𝑓)→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 𝑇𝐵={5,10,15,20} was selected. This support corresponds to the set of nonsupport zeroes of the cocontrol 𝑇𝑛0={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.

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).