On the Optimal Control Problem for Vibrations of the Rod/String Consisting of Two Non-Homogeneous Sections with the Condition at an Intermediate Time

: We consider an optimal boundary control problem for a one-dimensional wave equation consisting of two non-homogenous segments with piecewise constant characteristics. The wave equation describes the longitudinal vibrations of a non-homogeneous rod or the transverse vibrations of a non-homogeneous string with given initial, intermediate, and ﬁnal conditions. We assume that wave travel time for each of the sections is the same. The control is carried out by shifting one end with the other end ﬁxed. The quality criterion is set on the entire time interval. A constructive approach to building an optimal boundary control is proposed. The results obtained are illustrated with an analytical example.


Introduction
Many researchers pay attention to the study of control problems and optimal control problems for vibration processes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Modeling and control of dynamic systems with intermediate conditions is an actively developing direction in modern control theory. In particular, Refs. [2][3][4][5][6][7][8][9][10][11][12][13][14][15] address the study of such problems. This scientific direction has not yet been sufficiently studied, is in the process of formation, and there are only some results on it. The study of problems for such heterogeneous distributed systems is provided, in particular, in [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The conditions that determine the contact interactions of materials of heterogeneous bodies are of great importance. Therefore, in the course of mathematical modeling, taking into account these conditions of conjugation (joint, gluing) of two sections with different physical characteristics of materials should correspond to the conditions for the continuous outflow of excited wave processes. One of the first control problems for a distributed oscillatory system consisting of two piecewise homogeneous media was set by A.G. Butkovsky and studied in [8]. The problems of the optimization of the boundary control of vibrations of a rod consisting of heterogeneous sections were studied in [9,10] (and other works by the same author and his followers). For the study of these problems, the d'Alembert method was used. The authors of [13][14][15][16][17][18][19][20] studied boundary value problems for an equation describing the process of longitudinal vibrations of a rod with piecewise constant characteristics (consisting of at least two sections) with a free or fixed right end. Research was carried out in the class of generalized solutions.
This work aims to develop a constructive approach to building an optimal boundary control function for an inhomogeneous wave equation consisting of two heterogeneous sections with given initial, intermediate, and final conditions with a quality criterion given over the entire time interval.

Problem Statement
We consider longitudinal vibrations of a piecewise homogeneous rod located along the segment −l 1 ≤ x ≤ l and comprising two subsegments. The segment −l 1 ≤ x ≤ 0 has a uniform density ρ 1 = const, Young's modulus k 1 = const and a wave velocity The second segment 0 ≤ x ≤ l has a uniform density ρ 2 = const, Young's modulus k 2 = const and a wave velocity a 2 = k 2 ρ 2 . As in [9], we assumed that the lengths l 1 and l of the rod segments are such that the wave velocity on −l 1 ≤ x ≤ 0 coincides with the wave velocity on 0 ≤ x ≤ l, i.e., Let the state (longitudinal vibrations) of the rod (or transverse vibrations of the string) be described by the function Q (x, t), −l 1 ≤ x ≤ l, 0 ≤ t ≤ T, and the deviations from the equilibrium state satisfy the following wave equation, with the boundary conditions and with the conjugation conditions at the connection point x = 0 of the segments, The dynamics of a piecewise-homogeneous vibratory process represented by a homogeneous wave equation of variable structure (2) describes not only longitudinal vibrations of a piecewise-homogeneous rod (ρ is density, k is elasticity modulus), but also transverse vibrations of a piecewise-homogeneous string (ρ is density, k is string tension). Let there be given initial (for t = t 0 = 0) and final (for t = T) conditions, Additionally, let there be given at some intermediate moment In conditions (3), functions µ(t) are control actions (boundary control). It is assumed that Q(x, t) ∈ C 2 (Ω T ), where We also assume that for all functions the following consistency conditions are satisfied, Let us formulate the following problem of optimal boundary control of oscillations for system (2) with given values at intermediate times.
Among the possible controls µ(t), 0 ≤ t ≤ T, condition (3) is required to find such an optimal control that provides transition of the oscillatory motion of system (2) from a given initial state (5) to the final state (6), at the same time ensuring the fulfillment of condition (7) and minimizing the functional

Reduction to the Problem with Zero Boundary Conditions
To solve the problem under study, introduce a new variable [20], which leads to stretching or compression of the segment −l 1 ≤ x ≤ 0 with respect to the point x = 0. Taking into account (1), the segment −l 1 ≤ x ≤ 0 turns into the segment −l ≤ ξ ≤ 0. For the function Q(ξ, t), we obtain the same equations for the segments of the same length with the corresponding initial conditions boundary conditions intermediate conditions final conditions and conjugation conditions at the point ξ = 0 where the segments connect Since conditions (13) are not homogeneous, the solution to (11) can be constructed as a sum, where V(ξ, t) is a function with boundary conditions, that require definitions, whereas W(ξ, t) is a solution to (11) with non-homogeneous boundary conditions, The function W(ξ, t) has the form Substitute (17) into (11). Taking into account (20), we obtain the following equations for defining the function V(ξ, t): where The function V(ξ, t) satisfies the conjugation condition corresponding to (16) at the connection point ξ = 0 of the segments. Note that, according to (10), we have By virtue of conditions (12), (14), and (15), the function V(ξ, t) should satisfy the following set of conditions: initial and final Taking into account conditions (7) and (23), conditions (24)-(26) can be written as follows, respectively: Thus, the original problem has been reduced to the problem of motion optimal control described by Equation (21) with homogeneous boundary conditions (18), which is formulated as follows: it is required to find such optimal boundary control µ(t), 0 ≤ t ≤ T, that provides a transition of the oscillation described by Equation (21)

Reduction of the Problem with Zero Boundary Conditions to the Problem of Moments
Considering that the boundary conditions (18) are homogeneous, the consistency conditions are satisfied and the functions used belong to the indicated corresponding spaces, we seek the solution to (21) in the form: Use the Fourier series with the basis sin πkξ (21), (22) and conditions (27)-(29). We obtain Here, the Fourier coefficients of the functions F(ξ, t), ϕ i (ξ) (i = 0, 2), ψ 0 (ξ), and ψ T (ξ) are denoted by F k (t), ϕ Further, taking into account the intermediate (34) and final (35) conditions, we apply the approaches given in [2][3][4] to (36). Then, the control functions µ(t) for each k should satisfy the following integral relations: where h (1) Thus, the solution of the optimal control problem under study is reduced to finding such boundary controls µ(t), 0 ≤ t ≤ T, that for each k = 1, 2, . . . satisfy the integral relations (37) and provide a minimum to the functional (9). The optimal control problem for the functional (9) with integral conditions (38) can be considered as a conditional extremum problem from the calculus of variations.

Problem Solution
Since the functional (9) is the square of the norm of a linear normed space and the integral relations (37) generated by the functions µ(t) are linear, the problem of determining the optimal control for each k = 1, 2, . . . can be considered as a problem of moments [1,21]. Therefore, the solution can be constructed using the algorithm for solving the problem of moments.
In practice, it is common to select the first few n harmonics of elastic oscillations and solve the problem of control synthesis using methods of control theory for finitedimensional systems. Therefore, we construct a solution to problems (9) and (37) for k = 1, n using the algorithm for solving the problem of moments. Following [21], to solve the finite-dimensional (for k = 1, n) problem of moments (9) and (37), it is necessary to find p k , q k , γ k , k = 1, n, linked by the condition for which where The notation in Equation (40) means that the minimum of the functional is calculated by condition (39).
Without giving further constructions of the solution (since they are similar to the constructions given in [4,5]), note that the optimal boundary control µ 0 n (τ) for any n = 1, 2, . . . can be represented as: where Here the values p 0 k , q 0 k , γ 0 k , k = 1, n, are the solution to Equation (40) by condition (39), whereas It should be highlighted that the values of the optimal control µ 0 n (τ) at the end of the interval [0, t 1 ] coincide with the values at the beginning of the interval (t 1 , T], and this value has the following form: Therefore, the obtained optimal boundary controls µ 0 n (τ) are continuous on [0, T] as functions with respect to time.
Substituting the resulting expression for the optimal function µ 0 n (τ) into (32) and the expression that we found for F 0 k (t)-into (36), we obtain the function V 0 k (t), t ∈ [0, T]. Further, (30) entails that and, using (17) and (20), the optimal vibration function Q 0 n (ξ, t) for the first n harmonics will have the form where Taking into account notations from (10), and following (43)-(45), the optimal function Q 0 n (x, t) for −l 1 ≤ x ≤ l can be represented as The function Q 0 n (x, t) is continuous and it can be verified that it satisfies the conjugation condition at the point x = 0 where the segments (5) meet.

Constructing a Solution for n = 1
Applying the approach proposed above, construct an optimal boundary control for n = 1 (k = 1) and the corresponding string deflection function.
According to (36), we obtain: Following (43)-(45), the optimal state function Q 0 n (ξ, t) will have the form: According to (46), the explicit expression of the optimal function Q 0 1 (x, t) for −l 1 ≤ x ≤ l can be written as

The Example with Numerical Experiment
Let n = 1, a 1 = 1 4 , a 2 = 1 3 , l = 1, l 1 = 3 4 , t 0 = 0, t 1 = 3 l a 2 = 9, T = 6 l a 2 = 18, λ 1 = π 3 . Let the following initiate state be set for t = 0 the intermediate state for t = t 1 be given as and the trivial final states be defined for t = T as From formula (47) we will have the following system of algebraic equations where C 11 (T) = 273 32π 3 . The solution is so that The function µ 0 1 (t) has the form The graphical view of the function V 0 1 (t) is illustrated in Figure 1. From Q 0 1 (x, t) for −l 1 ≤ x ≤ l, we obtain at 0 ≤ t ≤ t 1 : at t 1 ≤ t ≤ T: Substituting t = 0, 9, 18 into (48), (49), we obtain the following relationship: Let us present the results of a comparative analysis based on residual We obtained: Graphical representations of the functions Q 0 1 (x, 0), Q 0 1 (x, 9), are given in Explicit expressions for the optimal function of boundary control µ 0 1 (t) and the corresponding function of deflection of an inhomogeneous string Q 0 1 (x, t) are constructed for n = 1. The performed calculations and comparisons of the results showed that the behavior of the functions of deflection of an inhomogeneous string is quite close to the given initial functions.

Conclusions
In this paper, we considered the problem of optimal boundary control of a onedimensional wave equation describing transverse vibrations of a piecewise homogeneous string or longitudinal vibrations of a piecewise homogeneous rod. A constructive approach was proposed for building an optimal boundary control function for one-dimensional non-homogeneous oscillatory processes. In this case, the explicit expression of the optimal boundary control function is represented through the given initial and final functions of the deflection and velocities of the points of the distributed system. The results can be used when designing the optimal boundary control of non-homogeneous oscillation processes in physical and technological systems.