Translation, solving scheme, and implementation of a periodic and optimal impulsive state control problem
 Ying Song^{1},
 Yongzhen Pei^{1, 2}Email authorView ORCID ID profile,
 Miaomiao Chen^{1} and
 Meixia Zhu^{1}
https://doi.org/10.1186/s1366201815150
© The Author(s) 2018
Received: 1 November 2017
Accepted: 6 February 2018
Published: 15 March 2018
Abstract
The periodic solution of the impulsive state feedback controls (ISFC) has been investigated extensively in the last decades. However, if the ecosystem is exploited in a period mode, what strategies are implemented to optimize the cost function at the minimal cost? Firstly, under the hypothesis that the system has a periodic solution, an optimal problem of ISFC is transformed into a parameter optimization problem in an unspecified time with inequality constraints, and together with the constraint of the first arrival threshold. Secondly, the rescaled time and a constraint violation function are introduced to translate the above optimal problem to a parameter selection problem in a specified time with the unconstraint. Thirdly, gradients of the objective function on all parameters are given to compute the optimal value of the cost function. Finally, three examples involving the marine ecosystem, computer virus, and resource administration are illustrated to confirm the validity of our approaches.
Keywords
1 Introduction
The topic about impulsive state feedback controls (abbreviated as ISFC) has been investigated extensively in the last decades due to its potential applications in culturing microorganisms [1–3], pest integrated management [4–6], disease control [7, 8], fish harvesting [9–11], and wildlife management [12, 13]. For example, [1] proposed a bioprocess model with ISFC to acquire an equivalent stable output by the precise feeding. Ref. [4] explored the periodic solution of an entomopathogenic nematode invading the insect model with ISFC. Ref. [7] considered some vaccines into a disease by ISFC, and got the uniqueness of order one periodic solution (OOPS) by geometric method. Ref. [12] formulated a whiteheaded langur’s ISFC model with sparse effect and continuous delay to study the periodic and artificial releasing. On ISFC models, scholars often pay close attention to the qualitative analysis of OOPS. Ref. [11] proposed a phytoplankton–fish model with ISFC and then formulated an optimal control problem (OCP, for short) and strived to find the appropriate harvesting rates to maximize the cost function in an impulsive period. Here, the solvability of the system in one period provides convenience for solving the OCP by Lagrange multiplier. But for the complicated ecosystem whose analytical solution cannot be expressed explicitly, if it is exploited in a period mode, what period and strategies are implemented to optimize the cost function at the minimal cost? Furthermore, how to translate the OCP of the ISFC to a problem with parameter optimization in one period is interesting. So far, few researchers have paid attention to these tasks which are the focuses of our paper.
The utilizations of optimal control can be found almost in all applied science fields, such as fishery model [14], iatrochemistry [15], switching powers [16], astrovehicle controls [17], undersea vehicles [18], ecoepidemiology [19], and virus therapies [20]. The Pontryagin principle and the Hamilton equation are the main theoretical tools to solve continuous system control [21]. However, the hybrid optimization problems involving the pulse threshold and system parameters are still sufficiently challenging and worth exploring. The control parameter technique offers the feasibility for solving this problem [21]. Teo et al. described the detailed and basic theory of the control parameter method in [22]. Until now, many important results have been achieved in recent years. We will apply these theories together with the constraint transcription technique [23] to resolve the above issues.
The other components of this paper are as follows. In Section 2, an optimal problem of state impulse feedback control is transformed into a parameter optimization problem in an unspecified time with inequality constraint, and together with the constraint of the first arrival threshold. In Section 3 we derive the required gradient formulas and present an algorithm for solving the approximate OCP. In Section 4, we give three examples and numerical simulations. Finally, a conclusion is provided in Section 5.
2 Problem statement and translation
 (H1)
Assume that \({f_{i}}\), ϕ, and I are continuously differentiable.
 (H2)
Denote the Euclidean norm by \(\\cdot\\). Suppose that there exists a constraint \(k>0\) meeting \({f_{i}}(\hat{\boldsymbol{y}})\leq k(1+\\hat{\boldsymbol{y}}\)\) for all \(\hat{\boldsymbol{y}}\).
 (H3)Assume that for fixed \(\hat{\boldsymbol{y}}_{0}\), (2.1) and (2.2) have a unique OOPS \(\Gamma _{{A}\rightarrow{B}}\) possessing period T, where A and B are the terminal and initial points of OOPS, respectively. When the impulsive effect take places, the point A is mapped to B, namely$${A} \bigl(\hat{y}_{1}(T)+I_{1} \bigl(\hat{\boldsymbol{y}}(T),\boldsymbol{\beta} \bigr),\ldots,\hat {y}_{n}(T)+I_{n} \bigl(\hat{\boldsymbol{y}}(T), \boldsymbol{\beta} \bigr) \bigr) \rightarrow{B}(\hat {y}_{10},\ldots, \hat{y}_{n0}). $$
 (H4)\(\hat{\boldsymbol{y}}(t)\in\Omega\), where$$ \Omega= \bigl\{ \hat{\boldsymbol{y}}(t)\mid \phi \bigl(\hat{\boldsymbol{y}}(t),\boldsymbol{\delta} \bigr)\neq 0 \mbox{ for } t\in(0,T) \mbox{ and }\phi \bigl(\hat{\boldsymbol{y}}(T), \boldsymbol {\delta} \bigr)=0 \bigr\} . $$(2.5)
Remark
If T is not the first positive time at which the solution \(\hat{\boldsymbol{y}}\) of (2.1) intersects with the surface \(\phi(\hat{\boldsymbol{y}}(t),\boldsymbol{\delta})=0\), then there exists \(0<\breve{T}<T\) such that \(\phi(\hat{\boldsymbol{y}}(\breve{T}),\boldsymbol {\delta})=0\). This appears in contradiction to the definitions of (2.5).
 (H5)
Assume that \(\Phi_{i}\), Ψ, and \(\Psi^{*}\) are continuously differentiable.
 (H6)
\(\Theta_{0}\) is continuously differentiable.
 (H7)
Functions \(L_{0}\) are continuously differentiable concerning the component \(\hat{\boldsymbol{y}}\) for each \(t\in [0,\hat{T}]\). Additionally, there is a constraint \(l>0\) such that \({L_{0}}(\hat{\boldsymbol{y}})\leq l(1+\\hat{\boldsymbol{y}}\)\) for all \(\hat{\boldsymbol{y}}\).
3 Solving scheme
By Theorems in [25] and [26], the next result is valid.
Lemma 3.1
The OCP (\({P_{0}}\)) ⇔ the problem (\({P_{1}}\)).
Next, we recommend an exact penalty method to overcome the remaining difficulty that the constraints (3.3) define a disjoint feasible region. Such constraints are referred to as functional inequality or path constraints. The essential dilemma about these constraints lies in the innumerable restriction on the state variables in the time scale [21].
 (\({P_{2}}\)):

Optimize a combined parameter vector \((\boldsymbol {\delta},\boldsymbol{\beta},T)\in\Lambda\times\Theta\times(0,\hat{T})\) and the new decision variable \(\varepsilon\in[0,\varepsilon_{1}]\) to minimize the transformed equivalent cost function \(J_{2}(\boldsymbol{\delta },\boldsymbol{\beta},T,\varepsilon)\) subject to the dynamics given by (3.1) and (3.2) in the interval \((0,1)\).
According to the main convergence result of [29], when ε̄ and δ approach zero, the cost of \(J_{2}\) approaches the optimal cost \(J_{1}\) of problem (\(P_{2}\)).
Theorem 3.1
Theorem 3.2
Note that, instead of an initial condition, the costate systems (3.10) and (3.11) involve a terminal value. So, we must integrate them from \(s=1\) to \(s=0\). Furthermore, in view of equations (3.12)–(3.15), we address the algorithm about calculating \(J_{2}\) and its gradients as follows.
Algorithm 1
 (i)
 (ii)
Use \(\boldsymbol{y}(s)\) to compute \(J_{2}\).
 (iii)
Use \(\boldsymbol{y}(s)\) and λ to compute \(\frac{\partial J_{2}}{\partial T}\), \(\frac{\partial J_{2}}{\partial\boldsymbol{\delta}}\), and \(\frac{\partial J_{2}}{\partial\boldsymbol{\beta}}\) according to equations (3.12), (3.13), and (3.14).
In the above, the methodology proposed involves transforming the periodic optimal control problem into a standard optimal control problem, after which standard computational techniques can be applied. Similarly, the case of the first positive time T assured by (2.11) can be derived.
4 Application
In this section, three examples are given to implement the above theories and approaches; and furthermore, to verify the validity of our algorithm.
Example 4.1
(Phytoplankton–fish system)
Results of simulation 1
Set  Nonoptimal control  Optimal control 

1  \(T_{0}=1.5\), \(e_{10}=0.7\)  \(T^{*}=1.6789\), \(e_{1}^{*}=0.6823\) 
\(e_{20}=0.9\), \(\varepsilon_{0}=0.1\)  \(e_{2}^{*}=0.9075\), \(\varepsilon ^{*}=0.1\)  
\(J_{3}=7931.6\), \(H_{0}=11.23 \)  \(J_{3}^{*}=8140.94\), \(H^{*}=14.49 \)  
2  \(T_{0}=1.5\), \(e_{10}=0.4\)  \(T^{*}=1.8038\), \(e_{1}^{*}=0.4285\) 
\(e_{20}=0.8\), \(\varepsilon_{0}=0.1\)  \(e_{2}^{*}=0.8429\), \(\varepsilon ^{*}=0.1\)  
\(J_{3}=6291.7\), \(H_{0}=11 \)  \(J_{3}^{*}=7592.98\), \(H^{*}=16.44 \)  
3  \(T_{0}=1\), \(e_{10}=0.7\)  \(T^{*}=1.8364\), \(e_{1}^{*}=0.7127\) 
\(e_{20}=0.9\), \(\varepsilon_{0}=0.1\)  \(e_{2}^{*}=0.9665\), \(\varepsilon ^{*}=0.1\)  
\(J_{3}=5333.51\), \(H_{0}=13.64 \)  \(J_{3}^{*}=8041.65\), \(H^{*}=16.84 \) 
Example 4.2
(Computer virus propagation under media coverage)
Results of simulation 2
Set  Nonoptimal control  Optimal control 

1  T = 8, \(b_{2}=0.6\), \(\varepsilon_{0}=0.1 \)  \(T^{*}=6.5499\), \(b_{2}^{*}=0.6214\), \(\varepsilon^{*}=4.521e{}5\) 
\(e_{1}=0.8\), \(e_{2}=0.9 \)  \(e_{1}^{*}=0.1613\), \(e_{2}^{*}=0.4572\)  
\(J_{1}=69.4267\), H = 0.7347  \(J^{*}=1.36176\), \(H^{*}=0.6632\)  
2  T = 7, \(b_{2}=0.55\), \(\varepsilon_{0}=0.1 \)  \(T^{*}=6.1392\), \(b_{2}^{*}=0.6345\), \(\varepsilon^{*}=1.61e{}6\) 
\(e_{1}=0.7\), \(e_{2}=0.6 \)  \(e_{1}^{*}=0.1703\), \(e_{2}^{*}=0.4371\)  
\(J_{1}=53.4043\), H = 0.7005  \(J^{*}=0.7111\), \(H^{*}=0.6396\)  
3  T = 6, \(b_{2}=0.5\), \(\varepsilon_{0}=0.1 \)  \(T^{*}=5.9094\), \(b_{2}^{*}=0.5386\), \(\varepsilon^{*}=0.441e{}7\) 
\(e_{1}=0.2\), \(e_{2}=0.4 \)  \(e_{1}^{*}=0.1553\), \(e_{2}^{*}=0.4422\)  
\(J_{1}=32.5521\), H = 0.6578  \(J^{*}=0.6984\), \(H^{*}=0.6454\) 
Example 4.3
(Speciesfood system)
Thus we can change problem (\(P_{0}\)) into the next problem.
Denote \(\bar{\tau}=(\bar{\tau}_{1},\bar{\tau}_{2},\ldots,\bar{\tau }_{n})\). Then problem (\(P_{2}\)) is given as follows:
Results of simulation for \(n=5\)
Set  Nonoptimal control  Optimal control 

1  \(\tau_{1}=3.23\), \(\tau_{2}=1.98\), \(\tau_{3}=1.44 \)  \(\tau_{1}^{*}=2.99\), \(\tau_{2}^{*}=1.77\), \(\tau_{3}^{*}=1.26\) 
\(\tau_{4}=1.13\), \(\tau_{5}=0.94 \)  \(\tau_{4}^{*}=0.98\), \(\tau _{5}^{*}=0.81\)  
λ = 4.51, α = 0.8773, \(\varepsilon_{0}=0.1 \)  \(\lambda ^{*}=4.4\), \(\alpha^{*}=0.89\), \(\varepsilon^{*}=6.581e{}5\)  
\(J_{3}^{5}=18.2467\), \(x_{1}=0.49 \)  \(J_{3}^{5*}=14.1559\), \(x_{1}^{*}=0.6\)  
2  \(\tau_{1}=2.35\), \(\tau_{2}=1.27\), \(\tau_{3}=0.88 \)  \(\tau_{1}^{*}=2.34\), \(\tau_{2}^{*}=1.26\), \(\tau_{3}^{*}=0.876\) 
\(\tau_{4}=0.67\), \(\tau_{5}=0.54 \)  \(\tau_{4}^{*}=0.67\), \(\tau _{5}^{*}=0.54\)  
λ = 4, α = 0.9028, \(\varepsilon_{0}=0.1 \)  \(\lambda ^{*}=3.99\), \(\alpha^{*}=0.9029\), \(\varepsilon^{*}=71e{}5\)  
\(J_{3}^{5}=13.5705\), \(x_{1}=1 \)  \(J_{3}^{5*}=10.6311\), \(x_{1}^{*}=4\)  
3  \(\tau_{1}=4.77\), \(\tau_{2}=3.82\), \(\tau_{3}=3.2 \)  \(\tau_{1}^{*}=4.74\), \(\tau_{2}^{*}=3.80\), \(\tau_{3}^{*}=3.17\) 
\(\tau_{4}=2.76\), \(\tau_{5}=2.44 \)  \(\tau_{4}^{*}=2.73\), \(\tau _{5}^{*}=2.40\)  
λ = 4.8, α = 0.7724, \(\varepsilon_{0}=0.1 \)  \(\lambda ^{*}=4.79\), \(\alpha^{*}=0.7274\), \(\varepsilon^{*}=21e{}4\)  
\(J_{3}^{5}=24.2191\), \(x_{1}=0.2 \)  \(J_{3}^{5*}=21.30982\), \(x_{1}^{*}=0.21\) 
Results of simulation for \(n=1\)
Set  Nonoptimal control  Optimal control 

1  T = 3.23, λ = 4.51  \(T ^{*}=2.93\), \(\lambda ^{*}=4.37\) 
α = 0.5881, ε = 0.1  \(\alpha ^{*}=0.62\), \(\varepsilon^{*}=1.57e{}4\)  
\(J_{3}^{1}=3.65\), \(x_{1}=0.49\)  \(J_{3}^{1*}=2.77\), \(x_{1}^{*}=0.63\)  
2  T = 2.35, λ = 4  \(T ^{*}=2.31\), \(\lambda ^{*}=3.96\) 
α = 0.6501, ε = 0.1  \(\alpha ^{*}=0.65\), \(\varepsilon^{*}=9e{}5\)  
\(J_{3}^{1}=2.71\), \(x_{1}=1\)  \(J_{3}^{1*}=2.01\), \(x_{1}^{*}=1.04\)  
3  T = 4.77, λ = 4.8  \(T ^{*}=4.42\), \(\lambda ^{*}=4.76\) 
α = 0.2952, ε = 0.1  \(\alpha ^{*}=0.41\), \(\varepsilon^{*}=2.64e{}4\)  
\(J_{3}^{1}=4.94\), \(x_{1}=0.2\)  \(J_{3}^{1*}=4.05\), \(x_{1}^{*}=0.24\) 
In order to compare the two optimal control modes (namely \(n=1\) and \(n=5\)), we compute \(5J^{1*}_{3}\) in view of Table 4, which represents that the mode \(n=1\) is performed five times. Compared with \(J^{5*}_{3}\), which represents that the mode \(n=5\) is executed one time, it indicates that the cost function \(5J^{1*}_{3}\) is slightly less than \(J^{5*}_{3}\). This result implies that the control mode of frequent releasing food and frequent harvest species is superior to that of frequent releasing food and infrequent harvest species.
5 Discussions
The topic about ISFC has been investigated extensively in the last decades due to its potential applications. Many authors have made every endeavor to explore the periodic solution of various systems including population, ecology, chemostat, epidemic, and so on. However, if the system is exploited in a period mode, what strategies are implemented to achieve the objective of optimal management? So far, few researchers keep a watchful eye on this task which has been our focus in the above sections. In summary, our approaches are concluded by the following three procedures. (1) Under the hypothesis that the ISFC system has a periodic solution, an optimal problem of ISFC is transformed into a parameter optimization problem in an unspecified time with inequality constraint, and together with the constraint of the first arrival threshold. (2) The rescaled time and a constraint violation function are introduced to translate the above optimal problem to a parameter selection problem in a specified time with the unconstraint. (3) The gradients of the objective function on all of parameters are given to compute the optimal value of the cost function. Finally, three examples involving the marine ecosystem, computer virus control, and resource administration are illustrated to confirm the validity of our approaches. In these examples, the parameters of the impulse and system on continuous systems and a hybrid system are optimized respectively.
Despite some endeavors in this paper, it may be beneficial to investigate a wider variety of topics in the future: (1) The actual data is necessary to achieve effective state feedback impulsive control. (2) Exploring other means to solve the OCP of the state dependent impulsive systems. (3) Applying this method to other fields. (4) For the optimal problems of the hybrid system in Example 4.3, the number of impulsive effect happens is worth exploring.
Declarations
Acknowledgements
The authors thank the referees for their careful reading of the original manuscript and many valuable comments and suggestions that greatly improved the presentation of this paper.
Funding
This work was supported by the National Natural Science Foundation of China (11471243, 11501409).
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final draft.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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