Two-end observability of elastic vibrations in distributed and lumped parameter systems

Finding and implementing the numerical solution of an optimal control problem for oscillations in a coupled objects system

“…The main limitation of the approach presented in the paper is that in the numerical solution of the considered problem it is not possible to vary some initial data in a wide range, it is necessary to take into account the conditions for matching the initial and boundary conditions (6), which narrows the class of admissible controls, for example, the lower limit of the external force p(t) is chosen taking into account conditions p min =a(0), the initial state of the distributed system, that is, the function a(x) must be a solution to the equation as-a'+a=0, the conditions a(1)=y 0 , b(1)=y 1 must be satisfied.…”

“…process control problems, which are described by the boundary value problem (1)-( 3), where y(t) acts as a control function, and p(t) is a fixed function or equal to zero (in the case when the left end of the string is fixed). Problems of this type were considered in [6].…”

“…Note that the approach based on obtaining formulas for calculating the first variation of the functional seems to be promising not only for optimizing systems similar to (1), ( 5), but also for more general ones, considered, for example, in [13]. In this case, the original distributed system can be solved by any numerical method without going over to ordinary differential equations, as it was done in [6,15].…”

“…which are approximating analogues of matching conditions (6), the new control p k+1 (t) for values t j =jΔt, j=3, 4, …, m, m=T/Δt is calculated by the formula:…”

“…The optimization of oscillatory processes holds significant relevance across various fields, encompassing endeavors such as stabilizing ship motions, managing crane booms, and controlling gas flows in extensive pipelines or power lines [1][2][3][4]. Notably, recent studies [5][6][7][8] have broached the challenges of damping oscillations within systems described by a combination of wave equations and second-order ordinary differential equations.…”