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Asymptotic Solution of a Singularly Perturbed Optimal Problem with Integral Constraint

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Abstract

Using the so-called direct scheme method, an asymptotic expansion of n-th order to the solution of a class of singularly perturbed linear-quadratic optimal problem with integral constraint on control is constructed. The expansion contains three type functions. Two of them are boundary layer functions in the vicinities of two fixed end-points, and the remain is regular function. A numerical example is represented to illustrate the obtained results.

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Acknowledgements

The author thanks the anonymous referees for numerous critical and constructive remarks that helped to improve the text of the paper. This research has been done under the Research Project QG.21.03 “SOME PROBLEMS ON THE STABILITY AND CONTROL OF SINGULAR SYSTEMS” of Vietnam National University, Hanoi.

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Authors and Affiliations

  1. University of Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam

    Thi Hoai Nguyen

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  1. Thi Hoai Nguyen
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Correspondence to Thi Hoai Nguyen.

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Communicated by Aram Arutyunov.

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Appendices

Appendices

For convenient, we use the notation

$$\begin{aligned} N_i(z_1(\cdot ), z_2(\cdot , \varepsilon ), t)= & {} \langle x_1(\cdot ), [W(t,\varepsilon )x_2(\cdot ,\varepsilon )]_i \rangle + \langle u_1(\cdot ), [R(t,\varepsilon )u_2(\cdot ,\varepsilon )]_i \rangle \\ M_i(z_1(\cdot , \varepsilon ), z_2(\cdot , \varepsilon ), t, k)= & {} \langle [x_1(\cdot , \varepsilon )]_{k-i}, [W(t,\varepsilon )x_2(\cdot ,\varepsilon )]_i \rangle \\&+ \,\langle [u_1(\cdot , \varepsilon )]_{k-i}, [R(t,\varepsilon )u_2(\cdot ,\varepsilon )]_i \rangle . \end{aligned}$$

Appendix A: The Construction of \(J_{2n}\)

The term \(J_{2n}^*\) is

$$\begin{aligned} J_{2n}^*= & {} \overline{J}_{2n} + \int _0^T(\frac{1}{2}F(\overline{z}_n(t), \overline{z}_n(t), t) + \langle \overline{x}_n, [W^1(t,\varepsilon )\overline{x}(t,\varepsilon )]_n\rangle \\&+\,\langle \overline{u}_n, [R^1(t,\varepsilon )\overline{u}(t,\varepsilon )]_n\rangle )\,\mathrm{d}t + \Pi \overline{J}_{2n-1} + Q\overline{J}_{2n-1}. \end{aligned}$$

Here,

$$\begin{aligned} \overline{J}_{2n}&= \int _0^T\left( \sum _{i=0}^{n-1}N_i(\overline{z}_{2n-i} (t),\overline{z}(t,\varepsilon ), t)\right) \,\mathrm{d}t, \\ \Pi \overline{J}_{2n-1}&= \int _0^{+\infty }\left( \sum _{i=0}^{n-1}M_i(\overline{z}(\tau \varepsilon , \varepsilon ), \Pi z(\tau ,\varepsilon ), \tau \varepsilon , 2n-1)\right) \,\mathrm{d}\tau \\&\quad + \, \int _0^{+\infty }\left( \sum _{i=0}^{n-1}N_i(\Pi _{2n-1-i}z(\tau ), \overline{z}(\tau \varepsilon , \varepsilon ), \tau \varepsilon )\right) \,\mathrm{d}\tau \\&\quad + \,\int _0^{+\infty }\left( \sum _{i=0}^{n-1}N_i(\Pi _{2n-1-i}z(\tau ),\Pi z(\tau , \varepsilon ), \tau \varepsilon )\right) \,\mathrm{d}\tau , \end{aligned}$$

the term \(Q\overline{J}_{2n-1}\) is obtained from the formula for \(\Pi \overline{J}_{2n-1}\) replacing the interval of integration over \([0,+\infty ]\) by \((-\infty , 0]\), the symbol \(\Pi \) by Q and the arguments \(\tau \varepsilon \) and \(\tau \) by \(T + \sigma \varepsilon \) and \(\sigma \), respectively.

Here and further, after the symbol \(\sim \) we mean the construction of the considered expression after omitting known summands.

We use (3.27), (3.26), (3.5), the technique of integration by parts, to construct \(\overline{J}_{2n}\). It follows that

$$\begin{aligned} \overline{J}_{2n}\sim & {} \sum _{i=0}^{n-1}\langle \overline{\psi }_i(t), \overline{x}_{2n-1-i}(t)\rangle |_0^T - \int _0^T(\langle \overline{x}_n, [A^1(t,\varepsilon )'\overline{\psi }_{n-1}(t,\varepsilon )]_n \nonumber \\&+ \,\frac{\mathrm{d}\overline{\psi }_{n-1}}{\mathrm{d}t}\rangle +\langle \overline{u}_n, [B^1(t,\varepsilon )'\overline{\psi }_{n-1}(t,\varepsilon )]_n\rangle )\mathrm{d}t\nonumber \\&+ \,\sum _{i=0}^{n-1}\overline{\alpha }_i\int _0^T\left( \sum _{j=0}^{n-1-i} [K(t,\varepsilon )]_j[\overline{u}(t,\varepsilon )]_{2n-i-j}\right) \,\mathrm{d}t. \end{aligned}$$
(A.1)

Using (3.36), (3.35), (3.31), the property (3.2), Eq. (3.5) with appropriate times of differentiation at \(t = 0\), we obtain the construction for the first integral in \(\Pi \overline{J}_{2n-1}\)

$$\begin{aligned} -\sum _{i=0}^{n-1}\langle \overline{x}_{2n-1-i}(0), \Pi _{i}\psi (0)\rangle . \end{aligned}$$
(A.2)

Using relations (3.26), (3.27) at \(t = 0\), and their appropriate times of differentiation at \(t = 0\) (note that \(\overline{\alpha }_i\), \(i = 0, 1, ..., n-1,\) are constants), the technique of integration by parts, the property (3.2), we obtain the construction for the second integral in \(\Pi \overline{J}_{2n-1}\)

$$\begin{aligned}&- \sum _{i=0}^{n-1}\langle \overline{\psi }_i(0), \Pi _{2n-1-i}x(0)\rangle \nonumber \\&\quad +\, \sum _{i=0}^{n-1}\overline{\alpha }_i\int _0^{+\infty } \left( \sum _{j=0}^{n-1-i}[K(\tau \varepsilon , \varepsilon )]_j[\Pi u(\tau ,\varepsilon )]_{2n-i-j-1}\right) \,\mathrm{d}\tau . \end{aligned}$$
(A.3)

Using (3.35), (3.36), (3.31), the property (3.2), we get the construction for the third integral in \(\Pi \overline{J}_{2n-1}\)

$$\begin{aligned} -\sum _{i=0}^{n-1}\langle \Pi _i\psi (0), \Pi _{2n-1-i}x(0)\rangle . \end{aligned}$$
(A.4)

Analogously [see (A.2)–(A.4)], we obtain the construction for \(Q\overline{J}_{2n-1}\)

$$\begin{aligned} Q\overline{J}_{2n-1}\sim & {} \sum _{i=0}^{n-1}\langle \overline{x}_{2n-1-i}(T), Q_{i}\psi (0)\rangle \;+ \sum _{i=0}^{n-1}\langle \overline{\psi }_i(T), Q_{2n-1-i}x(0)\rangle \nonumber \\&+ \, \sum _{i=0}^{n-1}\overline{\alpha }_i\int _{-\infty }^{0} \left( \sum _{j=0}^{n-1-i}[K(T+\sigma \varepsilon , \varepsilon )]_j[Qu(\sigma ,\varepsilon )]_{2n-i-j-1}\right) \,\mathrm{d}\sigma \nonumber \\&+ \,\sum _{i=0}^{n-1}\langle Q_i\psi (0), Q_{2n-1-i}x(0)\rangle . \end{aligned}$$
(A.5)

In view of the form of coefficients \(E_{i}\), \(i \ge 0\), in (3.4)

$$\begin{aligned} E_{i}= & {} \int _0^T[K(t,\varepsilon )\overline{u}(t,\varepsilon )]_{i}\,\mathrm{d}t \nonumber \\&+\, \int _0^{+\infty }[K(\tau \varepsilon , \varepsilon )\Pi u(\tau ,\varepsilon )]_{i-1}\,\mathrm{d}\tau + \int _{-\infty }^0[K(T+\sigma \varepsilon , \varepsilon )Q u(\sigma ,\varepsilon )]_{i-1}\,\mathrm{d}\sigma ,\nonumber \\ \end{aligned}$$
(A.6)

and using the conditions (3.32), (3.37), from (A.1) to (A.5), after neglecting known summands, we obtain the construction for \(J_{2n}\) denoted by \(\overline{J}_{n}(\overline{u}_n)\), which has the form (3.24).

Appendix B: The Construction of \(J_{2n+1}\)

The term \(J_{2n+1}^*\) is

$$\begin{aligned} J_{2n+1}^*= & {} \overline{J}_{2n+1} + \int _0^{+\infty }\Bigg (\frac{1}{2}F(\Pi _nz(\tau ), \Pi _nz(\tau ),0) \\&+\, \langle \Pi _nx, [W^1(\tau \varepsilon , \varepsilon )\Pi x(\tau , \varepsilon )]_n \rangle + \langle \Pi _nu, [R^1(\tau \varepsilon , \varepsilon )\Pi u(\tau ,\varepsilon )]_n \rangle \Bigg )\,\mathrm{d}\tau \\&+\, \int _{-\infty }^0(\frac{1}{2}F(Q_nz(\sigma ), Q_nz(\sigma ), T) + \langle Q_nx, [W^1(T+\sigma \varepsilon , \varepsilon )Q x(\sigma ,\varepsilon )]_n \rangle \\&+\,\langle Q_nu, [R^1(T+\sigma \varepsilon , \varepsilon ) Q u(\sigma ,\varepsilon )]_n \rangle )d\sigma + \Pi \overline{J}_{2n} + Q\overline{J}_{2n}, \end{aligned}$$

where

$$\begin{aligned} \overline{J}_{2n+1}= & {} \int _0^T\left( \sum _{i=0}^{n} N_i(\overline{z}_{2n+1-i}(t), \overline{z}(t,\varepsilon ),t)\right) \,\mathrm{d}t,\\ \Pi \overline{J}_{2n}= & {} \int _0^{+\infty }\left( \sum _{i=0}^{n-1}M_i(\overline{z}(\tau \varepsilon , \varepsilon ), \Pi z(\tau ,\varepsilon ), \tau \varepsilon , 2n)\right) \,\mathrm{d}\tau \\&+\, \int _0^{+\infty }\left( \sum _{i=0}^{n}N_i(\Pi _{2n-i}z(\tau ), \overline{z}(\tau \varepsilon ,\varepsilon ), \tau \varepsilon )\right) \,\mathrm{d}\tau \\&+\, \int _0^{+\infty }\left( \sum _{i=0}^{n-1}N_i(\Pi _{2n-i}z(\tau ), \Pi z(\tau , \varepsilon ),\tau \varepsilon )\right) \,\mathrm{d}\tau , \end{aligned}$$

and the term \(Q\overline{J}_{2n}\) is obtained from the formula for \(\Pi \overline{J}_{2n}\) replacing the interval of integration over \([0,+\infty ]\) by \((-\infty , 0]\), the symbol \(\Pi \) by Q and the arguments \(\tau \varepsilon \) and \(\tau \) by \(T+\sigma \varepsilon \) and \(\sigma \), respectively.

Using (3.27), (3.26), (3.25), the technique of integration by parts, we obtain the construction for \(\overline{J}_{2n+1}\)

$$\begin{aligned} \overline{J}_{2n+1}\sim & {} \sum _{i=0}^{n}\langle \overline{\psi }_i(t), \overline{x}_{2n-i}(t)\rangle |_0^T \nonumber \\&+\,\sum _{i=0}^{n}\overline{\alpha }_i\int _0^T \left( \sum _{j=0}^{n-i}[K(t,\varepsilon )]_j[\overline{u}(t,\varepsilon )]_{2n+1-i-j}\right) \,\mathrm{d}t. \end{aligned}$$
(B.1)

Using (3.36), (3.35), (3.31), the technique of integration by parts, the condition (3.2), we have the construction for the first integral in \(\Pi \overline{J}_{2n}\)

$$\begin{aligned} -\sum _{i=0}^{n-1}\langle \overline{x}_{2n-i}(0), \Pi _i\psi (0)\rangle . \end{aligned}$$
(B.2)

Using (3.27), (3.26) at \(t = 0\) and equations that follow from (3.27), (3.26) with appropriate time of differentiation at \(t = 0\), the relations (3.28), (3.31), the technique of integration by parts and the property (3.2), we obtain the construction for the second integral in \(\Pi \overline{J}_{2n}\)

$$\begin{aligned}&-\sum _{i=0}^{n}\langle \overline{\psi }_i(0), \Pi _{2n-i}x(0)\rangle \nonumber \\&\quad +\, \sum _{i=0}^{n} \overline{\alpha }_i \int _0^{+\infty }\left( \sum _{j=0}^{n-i}[K(\tau \varepsilon , \varepsilon )]_j[\Pi u(\tau ,\varepsilon )]_{2n-i-j}\right) \,\mathrm{d}\tau . \end{aligned}$$
(B.3)

Using (3.36), (3.35), (3.31), the technique of integration by parts, the property (3.2), we obtain the construction for the last integral in \(\Pi \overline{J}_{2n}\)

$$\begin{aligned}&-\sum _{i=0}^{n-1}\langle \Pi _i\psi (0), \Pi _{2n-i}x(0)\rangle \nonumber \\&\quad -\,\int _0^{+\infty }\langle \Pi _nx, [A^1(\tau \varepsilon , \varepsilon ) '\Pi _{n-1}\psi (\tau ,\varepsilon )]_n \rangle \,\mathrm{d}\tau . \end{aligned}$$
(B.4)

Analogously [see (B.2)–(B.4)], we get the construction for \(Q\overline{J}_{2n}\)

$$\begin{aligned} Q\overline{J}_{2n}\sim & {} \sum _{i=0}^{n-1}\langle \overline{x}_{2n-i}(T), Q_i\psi (0)\rangle + \sum _{i=0}^{n}\langle \overline{\psi }_i(T), Q_{2n-i}x(0)\rangle \nonumber \\&+\, \sum _{i=0}^{n}\overline{\alpha }_i\int _{-\infty }^0 \left( \sum _{j=0}^{n-i}[K(T+\sigma \varepsilon , \varepsilon )]_j[Qu(\sigma , \varepsilon )]_{2n-i-j}\right) \,\mathrm{d}\sigma \nonumber \\&+ \, \sum _{k=0}^{n-1}\langle Q_k\psi (0), Q_{2n-k}x(0)\rangle \nonumber \\&- \,\int _{-\infty }^0\langle Q_nx, [A^1(T + \sigma \varepsilon , \varepsilon )'Q_{n-1}\psi (\sigma ,\varepsilon )]_n \rangle \,\mathrm{d}\sigma . \end{aligned}$$
(B.5)

Using the conditions (3.32), (3.37), the coefficients \(E_i\), \(i = 0, 1, ..., 2n+1\), in (A.6), from (B.1) to (B.5), after neglecting known summands, we obtain the construction for \(J_{2n+1}\) which is denoted as \(\Pi _nJ(\Pi _nu) + Q_nJ(Q_nu)\) where the first and second terms have the forms (3.29) and (3.30), respectively.

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Nguyen, T.H. Asymptotic Solution of a Singularly Perturbed Optimal Problem with Integral Constraint. J Optim Theory Appl 190, 931–950 (2021). https://doi.org/10.1007/s10957-021-01916-w

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