# A.A. Shaburov. Asymptotic expansion of a solution to a singularly perturbed optimal control problem with a convex integral performance index whose terminal part depends on slow variables only ... P. 280-289

We consider an optimal control problem with a convex integral performance index for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control
$$\left\{\begin{array}\dot{x}_{\varepsilon}=A_{11}x_{\varepsilon}+A_{12}y_{\varepsilon}+B_{1}u, & t\in[0,T], & \|u\|\leqslant 1,\\ \varepsilon\dot{y}_{\varepsilon}=A_{22}y_{\varepsilon}+B_{2}u, & x_{\varepsilon}(0)=x^{0}, & y_{\varepsilon}(0)=y^{0},\\ J(u)\mathop{:=}\nolimits \varphi(x_{\varepsilon}(T))+\displaystyle\int\limits_{0}^{T}\,\|u(t)\|^2\,dt\rightarrow\min,\end{array}\right.$$ where $x\in\mathbb{R}^{n}$, $y\in\mathbb{R}^{m}$, $u\in\mathbb{R}^{r}$, $A_{ij}$ and $B_{i}$ for $i,j=1,2$ are constant matrices of corresponding dimension, and the function $\varphi(\cdot)$ is continuously differentiable in $\mathbb{R}^{n}$, strictly convex, and cofinite in the sense of convex analysis.
In the general case, Pontryagin's maximum principle is applied as a necessary and sufficient optimality condition in this problem, and there exists a unique vector $l_\varepsilon$ that defines an optimal control by the formula $$u_\varepsilon(T-t)=\frac{C_\varepsilon^{*}(t)l_\varepsilon} {S\left(\|C_\varepsilon^{*}(t)l_\varepsilon\|\right)},$$ where $$C_\varepsilon(t)\mathop{:=}\nolimits e^{\mathcal{A}_\varepsilon t} \mathcal{B}_\varepsilon=e^{A_{11}t} B_1 + \varepsilon^{-1}\mathcal {W}_\varepsilon(t)B_2, \qquad S(\xi)\mathop{:=}\nolimits \left\{ \begin{array}{ll} 2, & 0\leqslant \xi\leqslant2,\\[1ex] \xi, & \xi>2. \end{array}\right.$$ The main difference of this problem from the author's previous papers is that the terminal part of the performance index depends on the slow variables only and the control system has a more general form. It is proved that, in the case of a finite number of points where the type of the control is changed, a power asymptotic expansion can be constructed for the initial vector $l_\varepsilon$ of the conjugate system that defines the type of the optimal control.

Keywords: optimal control, singularly perturbed problems, asymptotic expansion, small parameter.

The paper was received by the Editorial Office on November 19, 2017.

Aleksandr Aleksandrovich Shaburov, doctoral student, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: alexandershaburov@mail.ru.

REFERENCES

1.  Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mishchenko E.F. The mathematical theory of optimal processes, ed. L.W. Neustadt, N Y, London, Interscience Publ. John Wiley & Sons, Inc., 1962, 360 p. ISBN: 0470693819 . Original Russian text published in Matematicheskaya teoriya optimal’nykh protsessov, Moscow: Fizmatgiz Publ., 1961, 391 p.

2.   Krasovskii N.N. Teoriya upravleniya dvizheniem. Lineinye sistemy. [Theory of motion control. Linear systems]. Moscow: Nauka Publ., 1968, 476 p.

3.   Lee E.B., Markus L. Foundations of optimal control theory. N Y; London; Sydney: John Wiley and Sons, Inc., 1967, 576 p. Translated to Russian under the title Osnovy teorii optimal’nogo upravleniya, Moscow, Nauka Publ., 1972, 576 p. ISBN: 0471522635 .

4.   Vasil’eva A.B., Dmitriev M.G. Singular perturbations in optimal control problems. J. Soviet Math., 1986, vol. 34, no. 3, pp. 1579–1629. doi: 10.1007/BF01262406 .

5.   Kokotovic P.V., Haddad A.H. Controllability and time-optimal control of systems with slow and fast modes. IEEE Trans. Automat. Control, 1975, vol. 20, no. 1, pp. 111–113. doi: 10.1109/TAC.1975.1100852 .

6.   Dontchev A.L. Perturbations, approximations and sensitivity analisis of optimal control systems, Berlin; Heidelberg; N Y; Tokio: Springer-Verlag, 1983, 161 p. doi: 10.1007/BFb0043612. Translated to Russian under the title Sistemy optimal’nogo upravleniya: Vozmushcheniya, priblizheniya i analiz chuvstvitel’nosti, Moscow, Mir Publ., 1987, 156 p.

7.   Kalinin A.I., Semenov K.V. Asymptotic optimization method for linear singularly perturbed systems with multidimensional control. Comput. Math. Math. Phys., 2004, vol. 44, no. 3, pp. 407–417.

8.   Danilin A.R., Parysheva Yu.V. Asymptotics of the optimal cost functional in a linear optimal control problem Dokl. Math., 2009, vol. 80, no. 1, pp. 478–481. doi: 10.1134/S1064562409040073 .

9.   Danilin A.R., Kovrizhnykh O.O. Time-optimal control of a small mass point without environmental resistance. Dokl. Math., 2013, vol. 88, no. 1, pp. 465–467. doi: 10.1134/S1064562413040364 .

10.   Rockafellar R. Convex analysis. Princeton, Princeton University Press, 1970, 451 p. ISBN: 0691015864 . Translated to Russian under the title Vypuklyi analiz, Moscow: Mir Publ., 1973, 470 p.

11.   Shaburov A.A. Asymptotic expansion of a solution for one singularly perturbed optimal control problem in ℝn with a convex integral quality index. Ural Math. J., 2017, vol. 3, no. 1, pp. 65–75. doi: 10.15826/umj.2017.1.005 .

12.   Il’in A.M., Danilin A.R. Asimptoticheskie metody v analize. [Asymptotic methods in analysis]. Moscow: Fizmatlit Publ., 2009, 248 p. ISBN: 978-5-9221-1056-3 .

13.   Danilin A.R. Asymptotic behavior of the optimal cost functional for a rapidly stabilizing indirect control in the singular case. Comput. Math. Math. Phys., 2006, vol. 46, no. 12, pp. 2068–2079. doi: 10.1134/S0965542506120062 .