Skip to main content
Log in

Optimal Control Computation for Nonlinear Fractional Time-Delay Systems with State Inequality Constraints

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, a numerical method is developed for solving a class of delay fractional optimal control problems involving nonlinear time-delay systems and subject to state inequality constraints. The fractional derivatives in this class of problems are described in the sense of Caputo, and they can be of different orders. First, we propose a numerical integration scheme for the fractional time-delay system and prove that the convergence rate of the numerical solution to the exact one is of second order based on Taylor expansion and linear interpolation. This gives rise to a discrete-time optimal control problem. Then, we derive the gradient formulas of the cost and constraint functions with respect to the decision variables and present a gradient computation procedure. On this basis, a gradient-based optimization algorithm is developed to solve the resulting discrete-time optimal control problem. Finally, several example problems are solved to demonstrate the effectiveness of the developed solution approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 78, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.P.: A formulation and numerical scheme for fractional optimal control problems. J. Vib. Control 14, 1291–1299 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alipour, M., Rostamy, D., Baleanu, D.: Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control 19, 2523–2540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alizadeh, A., Effati, S.: An iterative approach for solving fractional optimal control problems. J. Vib. Control 24, 18–36 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)

    Article  MATH  Google Scholar 

  6. Baillie, R.T.: Long memory processes and fractional integration in econometrics. J. Econom. 75, 5–59 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  7. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012)

    Book  MATH  Google Scholar 

  8. Balochian, S., Rajaee, N.: Fractional-order optimal control of fractional-order linear vibration systems with time delay. Int. J. Syst. Dyn. Appl. 7, 72–93 (2018)

    Google Scholar 

  9. Betts, J.T., Campbell, S., Thompson, K.: Optimal control software for constrained nonlinear systems with delays. In: Proceedings of IEEE Multi-conference on Systems and Control, pp. 444–449, Denver, USA (2011)

  10. Bhrawy, A.H., Doha, E.H., Machado, J.A., Ezz-Eldien, S.S.: An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian J. Control 17, 2389–2402 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bhrawy, A.H., Ezz-Eldien, S.S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53, 521–543 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cong, N.D., Tuan, H.T.: Existence, uniqueness, and exponential boundedness of global solution to delay fractional differential equations. Mediterr. J. Math. 14, 193 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Debeljković, D.: Time-Delay Systems. InTech, Rijeka (2011)

    Book  Google Scholar 

  14. Dehghan, M., Hamedi, E., Khosravian-Arab, H.: A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. J. Vib. Control 22, 1547–1559 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ezz-Eldien, S.S., Doha, E.H., Baleanu, D., Bhrawy, A.H.: A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. J. Vib. Control 23, 16–30 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ghomanjani, F., Farahi, M.H., Gachpazan, M.: Optimal control of time-varying linear delay systems based on the Bezier curves. Comput. Appl. Math. 33, 687–715 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gong, Z., Liu, C., Teo, K.L., Wang, S., Wu, Y.: Numerical solution of free final time fractional optimal control problems. Appl. Math. Comput. 405, 126270 (2021)

    MathSciNet  Google Scholar 

  18. Guglielmi, N., Hairer, E.: Geometric proofs of numerical stability for delay equations. IMA J. Numer. Anal. 21, 439–450 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, T.L.: The necessary conditions of fractional optimal control in the sense of Caputo. J. Optim. Theory Appl. 156, 115–126 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hosseinpour, S., Nazemi, A.: A collocation method via block-pulse functions for solving delay fractional optimal control problems. IMA J. Math. Control Inf. 34, 1215–1237 (2016)

    MathSciNet  MATH  Google Scholar 

  21. Hosseinpour, S., Nazemi, A., Tohidi, E.: Müntz–Legendre spectral collocation method for solving delay fractional optimal control problems. J. Comput. Appl. Math. 351, 344–363 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kamocki, R.: On the existence of optimal solutions to fractional optimal control problems. Appl. Math. Comput. 235, 94–104 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Kamocki, R.: Pontryagin maximum principle for fractional ordinary optimal control problems. Math. Methods Appl. Sci. 37, 1668–1686 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kamocki, R., Majewski, M.: Fractional linear control systems with Caputo derivative and their optimization. Optim. Control Appl. Methods 36, 953–967 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kibass, A.A., Srivastava, A.M., Trujillo, I.I.: Theory and Application of Fractional Differential Equations. Elsevier, New York (2006)

    Google Scholar 

  26. Kumar, M.: Optimal design of fractional delay FIR filter using cuckoo search algorithm. Int. J. Circuit Theory Appl. 46, 2364–2379 (2018)

    Article  Google Scholar 

  27. Laforgia, A., Natalini, P.: Exponential, gamma and polygamma functions: simple proofs of classical and new inequalities. J. Math. Anal. Appl. 407, 459–504 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lenz, S.M., Schlöder, J.P., Bock, H.G.: Numerical computation of derivatives in systems of delay differential equations. Math. Comput. Simul. 96, 124–156 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, W., Wang, S., Rehbock, V.: A 2nd-order one-step numerical integration scheme for a fractional differential equation. Numer. Algebra Control Optim. 7, 273–287 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, W., Wang, S., Rehbock, V.: Numerical solution of fractional optimal control. J. Optim. Theory Appl. 180, 556–573 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Magin, R.L.: Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 1–104 (2004)

    Article  Google Scholar 

  32. Martin, R., Quintana, J.J., Ramos, A., de la Nuez, I.: Modeling electrochemical double layer capacitor, from classical to fractional impedance. In: Proceedings of the IEEE Mediterranean Electrotechnical Conference, pp. 61–66, Ajaccio, France (2008)

  33. Marzban, H.R.: Solution of a specific class of nonlinear fractional optimal control problems including multiple delays. Optim. Control Appl. Methods 42, 2–29 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  34. Marzban, H.R.: A new fractional orthogonal basis and its application in nonlinear delay fractional optimal control problems. ISA Trans. 114, 106–119 (2021)

    Article  Google Scholar 

  35. Marzban, H.R., Malakoutikhah, F.: Solution of delay fractional optimal control problems using a hybrid of block-pulse functions and orthonormal Taylor polynomials. J. Frankl. Inst. 356, 8182–8251 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  36. Marzban, H.R., Pirmoradian, H.: A direct approach for the solution of nonlinear optimal control problems with multiple delays subject to mixed state-control constraints. Appl. Math. Model. 53, 189–213 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mohammadzadeh, R., Lakestani, M.: Optimal control of linear time-delay systems by a hybrid of block-pulse functions and biorthogonal cubic Hermite spline multiwavelets. Optim. Control Appl. Methods 39, 357–376 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  38. Moradi, L., Mohammadi, F., Baleanu, D.: A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J. Vib. Control 25, 310–324 (2019)

    Article  MathSciNet  Google Scholar 

  39. Mu, P., Wang, L., Liu, C.: A control parameterization method to solve the fractional-order optimal control problem. J. Optim. Theory Appl. 187, 234–247 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  40. Nazemi, A., Mansoori, M.: Solving optimal control problems of the time-delayed systems by Haar wavelet. J. Vib. Control 22, 2657–2670 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  42. Pooseh, S., Almieda, R., Torres, D.F.M.C.: Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10, 363–381 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rahimkhani, P., Ordokhani, Y., Babolian, E.: An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dyn. 86, 1649–1661 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. Safaie, E., Farahi, M.H., Ardehaie, M.F.: An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials. Comput. Appl. Math. 34, 831–846 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Salati, A.B., Shamsi, M., Torres, D.F.M.: Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 67, 334–350 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  46. Sweilam, N.H., AL-Mekhlafi, S.M.: Optimal control for a time delay multi-strain tuberculosis fractional model: a numerical approach. IMA J. Math. Control Inf. 36, 317–340 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tang, X., Liu, Z., Wang, X.: Integral fractional pseudospectral methods for solving fractional optimal control problems. Automatica 62, 304–311 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tang, X., Shi, Y., Wang, L.L.: A new framework for solving fractional optimal control problems using fractional pseudospectral method. Automatica 78, 333–340 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Teo, K.L., Li, B., Yu, C., Rehbock, V.: Applied and Computational Optimal Control. Springer, Berlin (2021)

    Book  MATH  Google Scholar 

  50. Tricaud, C., Chen, Y.: An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59, 1644–1655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tyrtyshnikov, E.E.: A Brief Introduction to Numerical Analysis. Birkhäuser, Boston (1997)

    Book  MATH  Google Scholar 

  52. Wang, Z., Hong, X., Shi, G.: Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62, 1531–1539 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yu, C., Lin, Q., Loxton, R., Teo, K.L., Wang, G.: A hybrid time-scaling transformation for time-delay optimal control problems. J. Optim. Theory Appl. 169, 876–901 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11771008), the Australian Research Council (No. DP190103361), the China Scholarship Council (No. 201902575002), and the Natural Science Foundation of Shandong Province, China (Nos. ZR2017MA005 and ZR2019MA031).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Chongyang Liu.

Additional information

Communicated by Ebrahim Sarabi.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proof of Theorem 3.1

We will prove the theorem in the following steps.

1.1 Step 1: Preliminaries

For \(k=1,\ldots ,N\), define \({\tilde{C}}^k=({\tilde{c}}_1^k,\ldots ,{\tilde{c}}_n^k)^\top \), where

$$\begin{aligned} {\tilde{c}}_i^k&=x^0_i+{\bar{h}}_i\sum \limits _{l=1}^{k-1}\big [f_i(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl_\tau },u^i_{kl},{\tilde{u}}^i_{kl_\sigma })\cdot ((k-l+1)^{\alpha _i}-(k-l)^{\alpha _i})\big ]\\&\quad +{\bar{h}}_if_i(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau }){\tilde{x}}^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })\\&\quad -\sum \limits _{p=1}^nb^k_{ip}x_p(t_{k-1})+{\mathcal {O}}(h^2). \end{aligned}$$

Note that this vector can be regarded as a disturbance of vector \(C^k\) defined in (22). Furthermore, by using linear interpolation, \(x(s^i_{kl})\), \(x(s^i_{kl}-\tau )\), \(u(s^i_{kl})\) and \(u(s^i_{kl}-\sigma )\) in (13) can be, respectively, expressed as:

$$\begin{aligned}&x_p(s^i_{kl})=x_p(t_{l-1})+\rho ^i_{kl}(x_p(t_{l})-x_p(t_{l-1}))+{\mathcal {O}}(h^2), \quad p=1,\ldots ,n, \end{aligned}$$
(48)
$$\begin{aligned}&x_p(s^i_{kl}-\tau )\nonumber \\&={\left\{ \begin{array}{ll}\phi _p(s^i_{kl}-\tau ), &{} \text{ if } s^i_{kl}-\tau \le 0,\\ x_p(t_{l_\tau -1})+{\bar{\rho }}^i_{kl_\tau }(x_p(t_{l_\tau })-x_p(t_{l_\tau -1}))+{\mathcal {O}}(h^2), &{} \text{ if } s^i_{kl}-\tau \in ]t_{l_\tau -1},t_{l_\tau }], \end{array}\right. }\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad ~~~~~~~~~~~~~p=1,\ldots ,n, \end{aligned}$$
(49)
$$\begin{aligned}&u(s^i_{kl})=u^i_{kl}+1^r{\mathcal {O}}(h^2), \end{aligned}$$
(50)

and

$$\begin{aligned} u(s^i_{kl}-\sigma )={\tilde{u}}^i_{kl_\sigma }+\delta ^i_{kl_\sigma }1^r{\mathcal {O}}(h^2), \end{aligned}$$
(51)

for all feasible i, k and l, where \(u^i_{kl}\) and \({\tilde{u}}^i_{kl_\sigma }\) are as defined in (14) and (16), respectively; and \(1^r\) is a column vector of all ones in \({\mathbb {R}}^r\).

1.2 Step 2: The Truncation Error \(R^i_k\) is of Order \({\mathcal {O}}(h^2)\)

Let \(i\in \{1,\ldots ,n\}\), \(k\in \{1,\ldots ,N\}\) and \(l\in \{1,\ldots ,k\}\). From (11), we have

$$\begin{aligned} |R^i_{kl}|&=\bigg |\frac{\displaystyle c^i_{kl}}{\displaystyle \varGamma (\alpha _i)}\int ^{lh}_{(l-1)h}(kh-s)^{\alpha _i-1}(s-s^i_{kl})\mathrm{d}s\bigg |\\&\le \frac{\displaystyle |c^i_{kl}|h^2}{\displaystyle \varGamma (\alpha _i)}\int _{(l-1)h}^{lh}(kh-s)^{\alpha _i-1}\mathrm{d}s\\&=\frac{\displaystyle |c^i_{kl}|h^2}{\displaystyle \varGamma (\alpha _i+1)}\big \{[(k-l+1)h]^{\alpha _i}-[(k-l)h]^{\alpha _i}\big \}. \end{aligned}$$

By Assumption 2.1, f is twice continuously differentiable. Thus, \(c^i_{kl}\) is bounded on \([-\tau ,T]\). Let \({\bar{c}}=\max \nolimits _{\begin{array}{c} i\in \{1,\ldots ,n\}\\ k\in \{1,\ldots ,N\} \end{array}}\max \nolimits _{l\ \in \{1,\ldots ,k\}}|c^i_{kl}|\). Then, from the definition of \(R^i_k\), we have

$$\begin{aligned} |R^i_k|&\le \sum \limits _{l=1}^{k}|R^i_{kl}|\\&\le \frac{\displaystyle {\bar{c}}h^2}{\displaystyle \varGamma (\alpha _i+1)}\sum \limits _{l=1}^{k} \big \{[(k-l+1)h]^{\alpha _i}-[(k-l)h]^{\alpha _i}\big \}\\&=\frac{\displaystyle {\bar{c}}h^2}{\displaystyle \varGamma (\alpha _i+1)}(kh)^{\alpha _i}\le \frac{\displaystyle {\bar{c}}h^2}{\displaystyle \varGamma (\alpha _i+1)}T^{\alpha _i}. \end{aligned}$$

Recall that \(\alpha _i\in ]0,1]\) for \(i=1,\ldots ,n\). Then, it follows from [27] that \(\varGamma (\alpha _i+1)\in [2^{\alpha _i-1},1]\). Also, it is obvious that \(T^{\alpha _i}\in ]1,T]\). Choosing \({\bar{C}}=2{\bar{c}}T\) gives

$$\begin{aligned} |R^i_k|\le {\bar{C}}h^2={\mathcal {O}}(h^2). \end{aligned}$$
(52)

1.3 Step 3: \(B^k\) is Non-singular

By Assumption 2.1, f is twice continuously differentiable in x(t) and \(x(t-\tau )\). Thus, \(\displaystyle \frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\) and \(\displaystyle \frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\) are bounded on [0, T] for \(i=1,\ldots ,n\). Let

$$\begin{aligned} M=\max \limits _{\begin{array}{c} i\in \{1,\ldots ,n\}\\ p\in \{1,\ldots ,n\} \end{array}}\bigg \{\left| \displaystyle \frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\right| ,\left| \displaystyle \frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\right| \bigg \}. \end{aligned}$$

Note that \(\rho ^i_{kk}\in ]0,1[\) and \({\bar{\rho }}^i_{kk_\tau }\in ]0,1]\) for all feasible i and k. It follows from (21) that

$$\begin{aligned} |b^k_{ip}|\le 2{\bar{h}}_iM=\frac{\displaystyle 2 h^{\alpha _i}M}{\displaystyle \varGamma (\alpha _i+1)}. \end{aligned}$$

Let \({\hat{h}}_i=\bigg (\displaystyle \frac{\displaystyle \varGamma (\alpha _i+1)}{\displaystyle 2nM}\bigg )^{\displaystyle \frac{\displaystyle 1}{\displaystyle \alpha _i}}\) and \({\hat{h}}=\min \limits _{i\in \{1,\ldots ,n\}}{\hat{h}}_i\). Then, for all \(h<{\hat{h}}\), \(\sum \limits _{p=1}^n|b^k_{ip}|<1,\ i=1,\ldots ,n.\) Since \(1-b^k_{ii}\ge 1-|b^k_{ii}|\), we have, for all \(i=1,\ldots ,n\),

$$\begin{aligned} \sum \limits _{\begin{array}{c} p=1\\ p\ne i \end{array}}^n|b^k_{ip}|<1-|b^k_{ii}|\le 1-b^k_{ii}, \end{aligned}$$

which implies that matrix \(B^k\) is strictly diagonally dominant. By the Levy–Desplanques theorem [51], \(B^k\) is non-singular.

1.4 Step 4: Base Case for (23)

For \(k=1\), it follows from (13) that, for \(i=1,\ldots ,n\),

$$\begin{aligned} x_i(t_1)=x^0_i+{\bar{h}}_i f_i(s^i_{11},x(s^i_{11}),x(s^i_{11}-\tau ),u(s^i_{11}),u(s^i_{11}-\sigma ))+R^i_1. \end{aligned}$$
(53)

Substituting (49)–(52) with \(k=l=1\) in (54) and applying Taylor expansion yield

$$\begin{aligned} x_i(t_1)&=\phi _i(0)+{\bar{h}}_if_i(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. \Big [\rho ^i_{11}(x_p(t_1)-x^0_p)+{\mathcal {O}}(h^2)\Big ]\bigg \}\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. \delta ^i_{11_\tau }\Big [{\bar{\rho }}^i_{11_\tau }(x_p(t_1)-x^0_p)+{\mathcal {O}}(h^2)\Big ]\bigg \}\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial u_m}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. {\mathcal {O}}(h^2)\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{u}}_m}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. \delta ^i_{11_\sigma }{\mathcal {O}}(h^2)+R^i_1. \end{aligned}$$
(54)

Recall that W is compact. By Assumption 2.1 and (53), (55) can be simplified as

$$\begin{aligned} x_i(t_1)&=\phi _i(0)+{\bar{h}}_if_i(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })\nonumber \\&\quad +\sum \limits _{p=1}^n\bigg [\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. {\bar{h}}_i\rho ^i_{11}(x_p(t_1)-x^0_p)\bigg ]\nonumber \\&\quad +\sum \limits _{p=1}^n\bigg [\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{11},x^{0},\delta ^i_{11_\tau }x^0+(1-\delta ^i_{11_\tau })\phi (s^i_{11}-\tau ),u^i_{11},{\tilde{u}}^i_{11_\sigma })}\right. {\bar{h}}_i\delta ^i_{11_\tau }{\bar{\rho }}^i_{11_\tau }(x_p(t_1)-x^0_p)\bigg ]\nonumber \\&\quad +{\mathcal {O}}(h^2). \end{aligned}$$
(55)

Thus, \(x(t_1)\) is a solution of the following linear equations:

$$\begin{aligned} B^1x(t_1)={\tilde{C}}^1. \end{aligned}$$
(56)

Since matrix \(B^1\) is non-singular when \(h<{\hat{h}}\) (recall that \({\hat{h}}\) is defined in Step 3), (20) as well as (57) has a unique solution. Thus, combining (57) with (20) yields

$$\begin{aligned} \Vert x(t_1)-x^1\Vert _\infty \le \Vert (B^1)^{-1}\Vert _\infty \Vert {\tilde{C}}^1-C^1\Vert _\infty ={\mathcal {O}}(h^2). \end{aligned}$$
(57)

This proves that (23) holds for \(k=1\) when \(h<{\hat{h}}\).

1.5 Step 5: Inductive Step for (23)

We now consider the case of \(k\ge 2\). Assume that, for \(l=1,\ldots ,k-1\),

$$\begin{aligned} \Vert x(t_l)-x^l\Vert _\infty ={\mathcal {O}}(h^2), \end{aligned}$$
(58)

when \(h<{\hat{h}}\).

From (13) and Step 2, we have

$$\begin{aligned} x_i(t_k)&=\phi _i(0)+R^i_k\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{l=1}^{k-1}f_i(s^i_{kl},x(s^i_{kl}),x(s^i_{kl}-\tau ),u(s^i_{kl}),u(s^i_{kl}-\sigma ))\big [(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}\big ]\nonumber \\&\quad +{\bar{h}}_if_i(s^i_{kk},x(s^i_{kk}),x(s^i_{kk}-\tau ),u(s^i_{kk}),u(s^i_{kk}-\sigma ))\nonumber \\&=\phi _i(0)+A^i_{k-1}+D^i_k+{\mathcal {O}}(h^2), \end{aligned}$$
(59)

where

$$\begin{aligned} A^i_{k-1}={\bar{h}}_i\sum \limits _{l=1}^{k-1}f_i(s^i_{kl},x(s^i_{kl}),x(s^i_{kl}-\tau ),u(s^i_{kl}),u(s^i_{kl}-\sigma ))[(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}], \end{aligned}$$

and

$$\begin{aligned} D^i_k={\bar{h}}_if_i(s^i_{kk},x(s^i_{kk}),x(s^i_{kk}-\tau ),u(s^i_{kk}),u(s^i_{kk}-\sigma )). \end{aligned}$$

Using (49)–(52) and applying Taylor expansion give

$$\begin{aligned} D^i_k&={\bar{h}}_if_i(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \Big [\rho ^i_{kk}(x_p(t_{k})-x_p(t_{k-1}))+{\mathcal {O}}(h^2)\Big ]\bigg \}\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \Big [\delta ^i_{kk_\tau }{\bar{\rho }}^i_{kk_\tau }(x_p(t_{k})-x_p(t_{k-1}))+{\mathcal {O}}(h^2)\Big ]\bigg \}\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial u_m}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. {\mathcal {O}}(h^2)\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{u}}_m}\left| _{(s^i_{kk},x(t_k),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. {\mathcal {O}}(h^2). \end{aligned}$$
(60)

By Assumption 2.1, Step 2, and (10), we can simplify (61) as:

$$\begin{aligned} D^i_k&={\bar{h}}_i f_i(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg [\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \rho ^i_{kk}(x_p(t_{k})-x_p(t_{k-1}))\bigg ]\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \Big [\delta ^i_{kk_\tau }{\bar{\rho }}^i_{kl_\tau }(x_p(t_{k})-x_p(t_{k-1}))\Big ]\bigg \}+{\mathcal {O}}(h^{2+\alpha _i}). \end{aligned}$$
(61)

Furthermore, using a Taylor expansion, we obtain

$$\begin{aligned}&f_i(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })\nonumber \\&\quad =f_i(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau }){\tilde{x}}^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })+{\bar{R}}^i_k, \end{aligned}$$
(62)

where

$$\begin{aligned} {\bar{R}}^i_k&=\frac{\displaystyle \partial f_i}{\displaystyle \partial x}\left| _{(s^i_{kk},\xi ^{k-1},\delta ^i_{kk_\tau }\zeta ^{k-1}+(1-\delta ^i_{kk_\tau })\nu ^{k-1},u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. (x(t_{k-1})-x^{k-1})\nonumber \\&\quad +\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}}\left| _{(s^i_{kk},\xi ^{k-1},\delta ^i_{kk_\tau }\zeta ^{k-1}+(1-\delta ^i_{kk_\tau })\nu ^{k-1},u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \delta ^i_{kk_\tau }(x(t_{k-1})-x^{k-1})\nonumber \\&\quad +\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}}\left| _{(s^i_{kk},\xi ^{k-1},\delta ^i_{kk_\tau }\zeta ^{k-1}+(1-\delta ^i_{kk_\tau })\nu ^{k-1},u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. (1-\delta ^i_{kk_\tau })(x(s^i_{kk}-\tau )-{\tilde{x}}^i_{kk_\tau });\nonumber \end{aligned}$$
(63)

\(\xi ^{k-1}\) and \(\zeta ^{k-1}\) are two points between \(x(t_{k-1})\) and \(x^{k-1}\); and \(\nu ^{k-1}\) is a point between \(x(s^i_{kk}-\tau )\) and \({\tilde{x}}^i_{kk_\tau }\). By Assumption 2.1 and (59), \(|{\bar{R}}^i_k|={\mathcal {O}}(h^2)\). Similarly, we have, for \(p=1,\ldots ,n\),

$$\begin{aligned}&\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad =\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. +{\mathcal {O}}(h^2), \end{aligned}$$
(64)

and

$$\begin{aligned}&\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x(t_{k-1}),\delta ^i_{kk_\tau }x(t_{k-1})+(1-\delta ^i_{kk_\tau })x(s^i_{kk}-\tau ),u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad =\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. +{\mathcal {O}}(h^2). \end{aligned}$$
(65)

Using estimations (63)–(65), (62) can be rewritten as:

$$\begin{aligned} D^i_k&={\bar{h}}_i f_i(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg [\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \rho ^i_{kk}(x_p(t_{k})-x_p(t_{k-1}))\bigg ]\nonumber \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \nonumber \\&\quad \times \Big [\delta ^i_{kk_\tau }{\bar{\rho }}^i_{kl_\tau }(x(t_{k})-x(t_{k-1}))\Big ]\bigg \}+{\mathcal {O}}(h^{2+\alpha _i}). \end{aligned}$$
(66)

Now, we consider the estimation for \(A^i_{k-1}\). By Assumption 2.1 and (49)–(51), we have

$$\begin{aligned} f_i(s^i_{kl},x(s^i_{kl}),x(s^i_{kl}-\tau ),u(s^i_{kl}),u(s^i_{kl}-\sigma )) =f_i(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})+{\tilde{R}}^i_{kl}, \end{aligned}$$

where

$$\begin{aligned} {\tilde{R}}^i_l&=\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})}\right. \Big [\big (x_p(t_{l-1}) \\&\quad +\rho ^i_{kl}(x_p(t_{l})-x_p(t_{l-1}))+{\mathcal {O}}(h^2)\big )-\big (x_p^{l-1}+\rho ^i_{kl}(x_p^{l}-x_p^{l-1})\big )\Big ]\bigg \} \\&\quad +\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})}\right. \theta ^{i}(l_\tau -1)\Big [\big (x_p(t_{l_\tau -1})+{\bar{\rho }}^i_{kl}(x_p(t_{l_\tau }) \\&\quad -x_p(t_{l_\tau -1}))+{\mathcal {O}}(h^2)\big )-\big (x_p^{l_\tau -1}+{\bar{\rho }}^i_{kl_\tau }(x_p^{l_\tau }-x_p^{l_\tau -1})\big )\Big ]\bigg \} \\&\quad +\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial u_m}\left| _{(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})}\right. {\mathcal {O}}(h^2)+\sum \limits _{m=1}^r\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{u}}_m}\left| _{(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})}\right. {\mathcal {O}}(h^2) \\&\quad +{\mathcal {O}}(h^2); \end{aligned}$$

and \(\theta ^i(\cdot )\) is as defined in (34). From (59) and Assumption 2.1, we obtain \(|{\tilde{R}}^i_{kl}|={\mathcal {O}}(h^2)\). Furthermore, using a similar proof as given for Step 2, we have

$$\begin{aligned} \displaystyle \sum \limits _{l=1}^{k-1}{\tilde{R}}^i_{kl}[(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}]={\mathcal {O}}(h^2). \end{aligned}$$

Thus,

$$\begin{aligned} A^i_{k-1}&={\bar{h}}_i\sum \limits _{l=1}^{k-1}f_i(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})[(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}]+{\tilde{R}}^i_k\nonumber \\&={\bar{h}}_i\sum \limits _{l=1}^{k-1}f_i(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})[(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}]+{\mathcal {O}}(h^2). \end{aligned}$$
(67)

Substituting (66) and (67) in (60) gives

$$\begin{aligned} x_i(t_k)&=\phi _i(0)+{\bar{h}}_if_i(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma }) \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg [\frac{\displaystyle \partial f_i}{\displaystyle \partial x_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \\&\quad \times \rho ^i_{kk}(x_p(t_{k})-x_p(t_{k-1}))\bigg ] \\&\quad +{\bar{h}}_i\sum \limits _{p=1}^n\bigg \{\frac{\displaystyle \partial f_i}{\displaystyle \partial {\tilde{x}}_p}\left| _{(s^i_{kk},x^{k-1},\delta ^i_{kk_\tau }x^{k-1}+(1-\delta ^i_{kk_\tau })x^i_{kk_\tau },u^i_{kk},{\tilde{u}}^i_{kk_\sigma })}\right. \\&\quad \times \Big [\delta ^i_{kk_\tau }{\bar{\rho }}^i_{kl_\tau }(x(t_{k})-x(t_{k-1}))\Big ]\bigg \}+{\mathcal {O}}(h^{2+\alpha _i}) \\&\quad +{\bar{h}}_i\sum \limits _{l=1}^{k-1}f_i(s^i_{kl},x^i_{kl},{\tilde{x}}^i_{kl},u^i_{kl},{\tilde{u}}^i_{kl})[(d_{kl}+1)^{\alpha _i}-d^{\alpha _i}_{kl}]+{\mathcal {O}}(h^2). \end{aligned}$$

Combining this equation with (20) and using (59), we have

$$\begin{aligned} \Vert x(t_k)-x^k\Vert _\infty \le \Vert B^k\Vert _\infty \Vert {\tilde{C}}^k-C^k\Vert _\infty ={\mathcal {O}}(h^2), \end{aligned}$$

from which we immediately infer that (58) holds for \(l=k\). \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Gong, Z., Yu, C. et al. Optimal Control Computation for Nonlinear Fractional Time-Delay Systems with State Inequality Constraints. J Optim Theory Appl 191, 83–117 (2021). https://doi.org/10.1007/s10957-021-01926-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-021-01926-8

Keywords

Mathematics Subject Classification

Navigation