# Abstract:

This note investigates the optimal control problem for a time-invariant linear systems with an arbitrary constant time-delay in in the input channel. A state feedback is provided for the infinite horizon case with a quadratic cost function. The solution is memoryless, except at an initial time interval of measure equal to the time-delay. If the initial input is set equal to zero, then the optimal feedback control law is memoryless from the beginning. Stability results are established for the closed loop system, in the scalar case.

# Keywords:

stability, optimal control, time-delay systems

93E12, 62A10

# References:

1. M. Basin: New Trends in optimal Filtering and Control for Polynomial and Time-Delay Systems. Lecture Notes in Control and Inform. Sci. 380, Springer-Verlag, Berlin - Heidelberg 2008.   CrossRef
2. M. Basin and R. Martanez-Zniga: Optimal linear filtering over observations with multiple delays. Internat. J. Robust Nonlinear Control 14 (2004), 8, 685-696.   CrossRef
3. M. Basin, J. Rodriguez-Gonzalez, L. Fridman and P. Acosta: Integral sliding mode design for robust filtering and control of linear stochastic time-delay systems. Internat. J. Robust Nonlinear Control 15 (2005), 9, 407-421.   CrossRef
4. M. Basin, J. Rodriguez-Gonzalez and R. Martinez-Zuniga: Optimal control for linear systems with time-delay in control input. J. Franklin Inst. - Engrg. and Appl. Math. 341 (2004), 3, 267-278.   CrossRef
5. E.-K. Boukas and Z.-K. Liu: Deterministic and Stochastic Time Delay Systems. Birkhauser, Boston 2002.   CrossRef
6. F. Carravetta, P. Palumbo and P. Pepe: Quadratic optimal control of linear systems with time-varying input delay. In: Proc. 49th IEEE Conf. on Dec. and Control (CDC), Atlanta 2010, pp. 4996-5000.   CrossRef
7. F. Carravetta, P. Palumbo and P. Pepe: Memoryless solution to the infinite horizon optimal control of LTI systems with delayed input In: Proc. IASTED Asian Conference on Modelling, Identification and Control (AsiaMIC), Phuket 2012.   CrossRef
8. Y. P. Chang, J. S. H. Tsai and L. S. Shieh: Optimal digital redesign of hybrid cascaded input-delay systems under state and control constraints. IEEE Trans. Circuits and Systems I - Fundamental Theory and Applications 49 (2002), 9, 1382-1392.   CrossRef
9. N. Chopra, P. Berestesky and M. W. Spong: Bilateral teleoperation over unreliable communication networks. IEEE Trans. Control Systems Technol. 16 (2008), 304-313.   CrossRef
10. D. H. Chyung: Discrete systems with delays in control. IEEE Trans. Automat. Control 14 (1969), 196-197.   CrossRef
11. M. C. Delfour: The linear quadratic optimal control problem with delays in state and control variables: A state space approach. SIAM J. Control Optim. 24 (1986), 835-883.   CrossRef
12. E. Fridman: New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Systems Control Lett. 43 (2001), 4, 309-319.   CrossRef
13. E. Fridman: Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl. 273 (2002), 24-44.   CrossRef
14. K. Gu, V. L. Kharitonov and J. Chen: Stability of Time Delay Systems. Birkhauser, Boston 2003.   CrossRef
15. A. Ichikawa: Quadratic control of evolution equations with delays in control. SIAM J. Control Optim. 5 (1982), 645-668.   CrossRef
16. A. Germani, C. Manes and P. Pepe: Implementation of an LQG control scheme for linear systems with delayed feedback action. In: Proc. 3rd European Control Conference (ECC), Vol. 4, Rome 1995, pp. 2886-2891.   CrossRef
17. A. Germani, C. Manes and P. Pepe: A twofold spline approximation for finite horizon LQG control of hereditary systems. SIAM J. Control Optim. 39 (2000), 4, 1233-1295.   CrossRef
18. Y. Kuang: Delay Differential Equations with Applications in Population Dynamics. Series Math. Sci. Engrg. 191, Academic Press, Boston 1993.   CrossRef
19. A. Kojima and S. Ishijima: Formulas on preview and delayed $H_\infty$ control. In: Proc. 42nd IEEE Conf. on Dec. and Control (CDC), Mauii 2003, pp. 6532-6538.   CrossRef
20. M. Krstic: Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Birkauser, Boston 2009.   CrossRef
21. Y. C. Ma, L. F. Huang and Q. L. Zhang: Robust guaranteed cost $H\sb \infty$ control for an uncertain time-varying delay system. (Chinese) Acta Phys. Sinica 56 (2007), 7, 3744-3752.   CrossRef
22. M. H. Milman: Approximating the linear quadratic optimal control law for hereditary systems with delays in the control. SIAM J. Control Optim. 2 (1988), 291-320.   CrossRef
23. A. A. Moelja and G. Meinsma: $H\sb 2$-optimal control of systems with multiple i/o delays: time domain approach. Automatica 41 (2005), 7, 1229-1238.   CrossRef
24. S. Mondié and W. Michiels: Finite spectrum assignment of unstable time-delay systems with a safe implementation. IEEE Trans. Automat. Control 48, 12, 2207-2212.   CrossRef
25. S.-I. Niculescu: Delay Effects on Stability, A Robust Control Approach. LNCIS 269, Springer-Verlag, London Limeted 2001.   CrossRef
26. Y. Kuang: Delay Differential Equations With Applications in Population Dynamics. Math. Sci. Engrg. 191, Academic Press Inc., San Diego 1993.   CrossRef
27. P. Pandolfi: The standard regulator problem for systems with input delays. An approach through singular control theory. Appl. Math. Optim. 31 (1995), 119-136.   CrossRef
28. P. Pepe and Z.-P. Jiang: A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems. Syst. Control Lett. 55 (2006), 1006-1014.   CrossRef
29. R. S. Pindyck: The discrete-time tracking problem with a time delay in the control. IEEE Trans. Automat. Control 17 (1972), 397-398.   CrossRef
30. I. Polushin, H. J. Marquez, A. Tayebi and P. X. Liu: A multichannel IOS small gain theorm for systems with multiple time-varying communication delays. IEEE Trans. Automat. Control 54 (2009), 404-409.   CrossRef
31. G. Tadmor: The standard $H_{\infty}$ problem in systems with a single input delay. IEEE Trans. Automat. Control 45 (2000), 382-397.   CrossRef
32. P. K. C. Wang: Optimal control for discrete-time systems with time-lag controls. IEEE Trans. Automat. Control 19 (1975), 425-426.   CrossRef
33. H. Zhang, G. Duan and L. Xie: Linear quadratic regulation for linear time-varying systems with multiple input delays. Automatica {\rm 42} (2006), 9, 1465-1476.   CrossRef
34. B. Zhou, Z. Lin and G.-R. Duan: Truncated predictor feedback for linear systems with long time-varying input delays. Automatica 48 (2012), 10, 2387-2389.   CrossRef