# American Institute of Mathematical Sciences

June  2018, 8(2): 211-235. doi: 10.3934/naco.2018013

## Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances

 1 Department of Applied Mathematics, ORT Braude College of Engineering, 51 Snunit Str., P.O.B. 78, Karmiel 2161002, Israel 2 Department of Mathematics, University of Haifa, Mount Carmel, Haifa 3498838, Israel

* Corresponding author

Received  March 2017 Revised  December 2017 Published  May 2018

An infinite horizon quadratic control of a linear system with known disturbance is considered. The feature of the problem is that the cost of some (but in general not all) control coordinates in the cost functional is much smaller than the costs of the other control coordinates and the state cost. Using the control optimality conditions, the solution of this problem is reduced to solution of a hybrid set of three equations, perturbed by a small parameter. One of these equations is a matrix algebraic Riccati equation, while two others are vector and scalar differential equations subject to terminal conditions at infinity. For this set of the equations, a zero-order asymptotic solution is constructed and justified. Using this asymptotic solution, a relation between solutions of the original problem and the problem, obtained from the original one by replacing the small control cost with zero, is established. Based on this relation, the best achievable performance in the original problem is derived. Illustrative examples are presented.

Citation: Valery Y. Glizer, Oleg Kelis. Asymptotic properties of an infinite horizon partial cheap control problem for linear systems with known disturbances. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 211-235. doi: 10.3934/naco.2018013
##### References:

show all references

##### References:
Optimal value of the cost functional (105)
 $\varepsilon$ 0.4 0.2 0.1 0.05 0.025 0.0125 0.00625 ${\mathcal J}_{\varepsilon}^{*}$ 10.8004 9.2653 8.6776 8.4775 8.4076 8.3809 8.3697
 $\varepsilon$ 0.4 0.2 0.1 0.05 0.025 0.0125 0.00625 ${\mathcal J}_{\varepsilon}^{*}$ 10.8004 9.2653 8.6776 8.4775 8.4076 8.3809 8.3697
 [1] Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97 [2] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [3] Fabio Bagagiolo. An infinite horizon optimal control problem for some switching systems. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 443-462. doi: 10.3934/dcdsb.2001.1.443 [4] Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879 [5] Yuan Tan, Qingyuan Cao, Lan Li, Tianshi Hu, Min Su. A chance-constrained stochastic model predictive control problem with disturbance feedback. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-13. doi: 10.3934/jimo.2019099 [6] Naïla Hayek. Infinite-horizon multiobjective optimal control problems for bounded processes. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1121-1141. doi: 10.3934/dcdss.2018064 [7] Vincenzo Basco, Piermarco Cannarsa, Hélène Frankowska. Necessary conditions for infinite horizon optimal control problems with state constraints. Mathematical Control & Related Fields, 2018, 8 (3&4) : 535-555. doi: 10.3934/mcrf.2018022 [8] Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131 [9] Gafurjan Ibragimov, Askar Rakhmanov, Idham Arif Alias, Mai Zurwatul Ahlam Mohd Jaffar. The soft landing problem for an infinite system of second order differential equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 89-94. doi: 10.3934/naco.2017005 [10] Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239 [11] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming formulations of deterministic infinite horizon optimal control problems in discrete time. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3821-3838. doi: 10.3934/dcdsb.2017192 [12] Alexander Tarasyev, Anastasia Usova. Application of a nonlinear stabilizer for localizing search of optimal trajectories in control problems with infinite horizon. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 389-406. doi: 10.3934/naco.2013.3.389 [13] Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control & Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013 [14] V.N. Malozemov, A.V. Omelchenko. On a discrete optimal control problem with an explicit solution. Journal of Industrial & Management Optimization, 2006, 2 (1) : 55-62. doi: 10.3934/jimo.2006.2.55 [15] Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 [16] Sergio Albeverio, Sonia Mazzucchi. Infinite dimensional integrals and partial differential equations for stochastic and quantum phenomena. Journal of Geometric Mechanics, 2019, 11 (2) : 123-137. doi: 10.3934/jgm.2019006 [17] Senda Ounaies, Jean-Marc Bonnisseau, Souhail Chebbi, Halil Mete Soner. Merton problem in an infinite horizon and a discrete time with frictions. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1323-1331. doi: 10.3934/jimo.2016.12.1323 [18] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [19] Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021 [20] Elena Goncharova, Maxim Staritsyn. On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1061-1070. doi: 10.3934/dcdss.2018061

Impact Factor:

## Metrics

• HTML views (105)
• Cited by (0)

• on AIMS