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.
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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).
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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
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:
and
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
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
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
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
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
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 \(i=1,\ldots ,n\),
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 \(i=1,\ldots ,n\),
Substituting (49)–(52) with \(k=l=1\) in (54) and applying Taylor expansion yield
Recall that W is compact. By Assumption 2.1 and (53), (55) can be simplified as
Thus, \(x(t_1)\) is a solution of the following linear equations:
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
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 \(l=1,\ldots ,k-1\),
when \(h<{\hat{h}}\).
From (13) and Step 2, we have
where
and
Using (49)–(52) and applying Taylor expansion give
By Assumption 2.1, Step 2, and (10), we can simplify (61) as:
Furthermore, using a Taylor expansion, we obtain
where
\(\xi ^{k-1}\) 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 \(p=1,\ldots ,n\),
and
Using estimations (63)–(65), (62) can be rewritten as:
Now, we consider the estimation for \(A^i_{k-1}\). By Assumption 2.1 and (49)–(51), we have
where
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
Thus,
Substituting (66) and (67) in (60) gives
Combining this equation with (20) and using (59), we have
from which we immediately infer that (58) holds for \(l=k\). \(\square \)
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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
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DOI: https://doi.org/10.1007/s10957-021-01926-8
Keywords
- Fractional time-delay system
- Fractional optimal control
- Inequality constraint
- Numerical integration
- Numerical optimization