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The Maximum Principle
Published in L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, K. N. Trirogoff, L. W. Neustadt, L. S. Pontryagin Selected Works, 2018
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, Κ. N. Trirogoff, L. W. Neustadt
theorem 3. Let u(t), t0 ≤ t ≤ t1, be an admissible control which transfers the phase point from some position x0 ϵ S0to the position x1 ϵ S1, and letx(t) be the corresponding trajectory (starting at the pointx0 = (0, x0))• In order that u(t) andx(t) yield the solution of the optimal problem with variable endpoints, it is necessary that there exist a nonzero continuous vector function ψ(t), which satisfies the conditions of Theorem 1, and in addition, the transversality condition at both endpoints of the trajectory x(t).
Observer path planning for maximum information
Published in Optimization, 2022
First we define the state variable vector for brevity. Note that . Then we define the Hamiltonian function for Problem in a standard manner as where is the adjoint (or costate) variable vector. In (5), we do not show the dependence of the variables on t, for clarity in appearance. Keeping up with the tradition we introduce the notation We also define the function such that The adjoint variable vector is assumed to satisfy the differential equation and the boundary condition (see e.g. [7]) where and . The boundary condition in (6) is referred to as the transversality condition. Componentwise, (6) can be re-written as Clearly, , for i = 1, 2, 3. Then (7d)–(7f) yield for all . After expanding the right-hand sides of the ODEs in (7a)–(7c), and carrying out manipulations, one gets
On construction of sampling patterns for preserving observability/controllability of linear sampled-data systems
Published in International Journal of Control, 2022
Ali Hamidoğlu, Elimhan N. Mahmudov
In the monograph Mahmudov (2011) and papers (Mahmudov, 2015a, 2015b, 2017, 2018a, 2018b, 2020; Aida-Zade & Abdullayev, 2020) consider different problems of optimal control theory with higher order linear/semilinear discrete and continuous systems (with fixed and varying time interval). Necessary and sufficient optimality conditions under the transversality condition are proved incorporating the Euler-Lagrange and Hamiltonian type inclusions/equations.
First-order necessary optimality conditions for infinite horizon optimal control problems with linear dynamics and convex objective
Published in Optimization, 2018
The problem under consideration here involves the Lebesgue integral in the objective and a linear dynamics. By vector-valued, we mean that the images of the state and control variables are n-dimensional and m-dimensional tuples, respectively. The importance of these problems is shown by the examples in [5]. The key idea of this paper is to discuss such problems in a Hilbert space setting. This has many advantages (see [6–8]) due to the properties of the corresponding spaces. So the key idea is to show that the trajectories, satisfying the state-equation and the initial condition, belong to a weighted Hilbert space due to a natural growth condition. This allows to introduce the weighted Sobolev space as the state space and the weighted Lebesgue space as the control space in the problem setting. This idea was already foreshadowed in [1] and used in the works [6–8]. However, in difference to [1] we use the more beneficial case of weighted Hilbert spaces here, and expand the ideas of [6–8] to systems of differential equations. Additionally, in difference to [1] we analyse the case of non-autonomous functions in the objective, as well. In this situation, we apply ideas from [6–8] and show a Pontryagin-type maximum principle by separation of convex, closed subsets in Hilbert spaces. Using this principle of proof, we do not need a locally Lipschitz-condition of the objective function deviating from the investigations in [1]. Instead, we use a less restrictive growth condition. By this separation approach, we also retrieve a transversality condition deviating from the investigations in [3]. This condition is particularly important for calculating numerical solutions, since we can transform the infinite horizon control problem to a two-point boundary value problem by the techniques presented in [9]. Without a transversality condition, we would have a loss of information from the optimal control problem replaced by the boundary value problem.