Abstract
In this manuscript, we study the finite-approximate controllability of impulsive fractional functional evolution equations of order \(1<\alpha <2\) in Hilbert spaces. We first discuss a useful characterization of the finite-approximate controllability for linear fractional evolution equations of order \(1<\alpha <2\) in terms of a resolvent-like operator. We also find a suitable control to obtain the approximate controllability of the linear system, which also ensures the finite-approximate controllability of the system. Next, we establish sufficient conditions for the finite-approximate controllability of the semilinear impulsive fractional evolution equations, whenever the corresponding linear system is approximately controllable. Moreover, we provide an example of fractional wave equation to illustrate the efficiency of the developed results. Finally, we discuss the finite-approximate controllability of semilinear fractional evolution equations of order \(1<\alpha <2\) with finite delay by using a variational method.
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References
Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems, 2nd edn. Birkhauser Verlag, New York (2011)
Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces. IMA J. Math. Control Inform. 39(4), 1103–11142 (2022)
Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of non-instantaneous impulsive fractional evolution equations of order \(1<\alpha <2\) with state-dependent delay in Banach spaces. Math. Methods Appl. Sci. 46(1), 531–559 (2023)
Arora, S., Mohan, M.T., Dabas, J.: Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces. J. Differ. Equ. 307, 83–113 (2022)
Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, New York (1993)
Barbu, V.: Controllability and Stabilization of Parabolic Equations. Birkhäuser, New York (2018)
Bashirov, A.E., Mahmudov, N.I.: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control. Optim. 37(6), 1808–1821 (1999)
Benchohra, M., Henderson, J., Ntouyas, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)
Chen, P., Zhang, X., Li, Y.: A blowup alternative result for fractional non-autonomous evolution equation of Volterra type. Commun. Pure Appl. Anal. 17(5), 1975–1992 (2018)
Chen, P., Zhang, X., Li, Y.: Approximate controllability of non-autonomous evolution system with nonlocal conditions. J. Dyn. Control Syst. 26(1), 1–16 (2020)
Chen, P., Zhang, X., Li, Y.: Cauchy problem for fractional non-autonomous evolution equations. Banach J. Math. Anal. 14(2), 559–584 (2020)
Chen, P., Zhang, X., Li, Y.: Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10(4), 955–973 (2019)
Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, New York (1996)
Grudzka, A., Rykaczewski, K.: On approximate controllability of functional impulsive evolution inclusions in a Hilbert space. J. Optim. Theory Appl. 166, 414–439 (2015)
He, J.W., Liang, Y., Ahmad, B., Zhou, Y.: Nonlocal fractional evolution inclusions of order \(\alpha \in (1,2)\). Mathematics 7(2), 1–17 (2019)
Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141(5), 1641–1649 (2013)
Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Kemppainen, J., Siljander, J., Vergara, V., Zacher, R.: Decay estimates for time fractional and other non-local in time subdiffusion equations in \({\mathbb{R} }^d\). Math. Ann. 366(3–4), 941–979 (2016)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)
Kisyński, J.: On cosine operator functions and one parameter group of operators. Studia Math. 44(1), 93–105 (1972)
Klafter, J., Metzler, R.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)
Klamka, J.: Controllability and Minimum Energy Control. Springer, Berlin (2018)
Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)
Lions, J.L., Zuazua, E.: The cost of controlling unstable systems: time irreversible systems. Rev. Mat. UCM 10(2), 481–523 (1997)
Li, X., Yong, J.: Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, New York (1995)
Mahmudov, N.I.: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control. Optim. 42(5), 1604–1622 (2003)
Mahmudov, N.I.: Finite-approximate controllability of evolution equations. Appl. Comput. Math. 16(2), 159–167 (2017)
Mahmudov, N.I.: Finite-approximate controllability of evolution systems via resolvent-like operators. https://arxiv.org/abs/1806.06930
Mahmudov, N.I.: Finite-approximate controllability of fractional evolution equations: variational approach. Fract. Calc. Appl. Anal. 21(4), 919–936 (2018)
Mahmudov, N.I.: Variational approach to finite-approximate controllability of Sobolev-Type fractional systems. J. Optim. Theory Appl. 184(2), 671–686 (2020)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press, London (2010)
Nesić, D., Teel, A.R.: Input-to-state stability of networked control systems. Automatica 40(12), 2121–2128 (2004)
Pazy, A.: Semigroup of Linear Operators and Applications to Partial Equations. Springer, Berlin (1983)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Raja, M.M., Vijayakumar, V.: New results concerning to approximate controllability of fractional integrodifferential evolution equations of order \(1<r<2\). Numer. Methods Partial Differ. Equ. 38(3), 509–524 (2022)
Raja, M.M., Vijayakumar, V., Udhayakumar, R., Zhou, Y.: A new approach on the approximate controllability of fractional differential evolution equations of order \(1 < r < 2\) in Hilbert spaces. Chaos Solitons Fract. 141, 110310 (2020)
Rossikhin, Y.A., Shitikova, M.V.: Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems. Acta Mech. 120(1–4), 109–125 (1997)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Philadelphia (1993)
Singh, S., Arora, S., Mohan, M.T., Dabas, J.: Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evol. Equ. Control Theory 11(1), 67–93 (2022)
Travis, C.C., Webb, G.: Cosine families and abstract nonlinear second order differential equations. Acta Math. Hungar. 32(1–2), 75–96 (1978)
Travis, C.C., Webb, G.: Second order differential equations in Banach space. In: Lakshmikantham, V. (ed) Nonlinear Equations in Abstract Spaces, pp. 331–361. Academic Press (1978)
Tavazoei, M.S., Haeri, M., Jafari, S., Bolouki, S., Siami, M.: Some applications of fractional calculus in suppression of chaotic oscillations. IEEE Trans. Commun. Electron. 55(11), 4094–4101 (2008)
Triggiani, R.: Addendum: a note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control. Optim. 18(1), 98 (1980)
Triggiani, R.: A note on the lack of exact controllability for mild solutions in Banach spaces. SIAM J. Control. Optim. 15(3), 407–411 (1977)
Tian, Y., Wang, J.R., Zhou, Y.: Almost periodic solutions for a class of non-instantaneous impulsive differential equations. Quaest. Math. 42(7), 885–905 (2019)
Vijayakumar, V., Ravichandran, C., Nisar, K.S., Kucche, K.D.: New discussion on approximate controllability results for fractional Sobolev type Volterra–Fredholm integrodifferential systems of order \(1<r <2\). Numer. Methods Partial Differ. Equ. https://doi.org/10.1002/num.22772 (2021)
Wang, J.R., Feĉkan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46(2), 915–933 (2015)
Wang, J.R., Feĉkan, M.: Non-instantaneous Impulsive Differential Equations. IOP Publishing, Berlin (2018)
Wang, J., Ibrahim, A.G., O’regan, D.: Finite-approximate controllability of Hilfer fractional semilinear differential equations. Miskolc Math. Notes 21(1), 489–507 (2020)
Wei, W., Xiang, X., Peng, Y.: Nonlinear impulsive integrodifferential equations of mixed type and optimal controls. Optimization 55(1–2), 141–156 (2006)
Yang, T., Chua, L.O.: Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 7(3), 645–664 (1997)
Yan, Z.: Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay. Int. J. Control 85(8), 1051–1062 (2012)
Zhou, Y., He, J.W.: New results on controllability of fractional evolution systems with order \(\alpha \in (1,2)\). Evol. Equ. Control Theory 10(3), 491–509 (2021)
Zuazua, E.: Chapter 7—controllability and observability of partial differential equations: some results and open problems. In: Dafermos, C.M., Feireisl, E. (eds) Handbook of Differential Equations: Evolutionary Equations, Vol. 3, pp. 527–621. North-Holland (2007)
Zuazua, E.: Controllability of Partial Differential Equations. 3rd cycle, Castro Urdiales (Espagne) (2006). https://cel.hal.science/cel-00392196
Zuazua, E.: Finite dimensional null controllability for the semilinear heat equation. J. Math. Pures Appl. 76(3), 570–594 (1997)
Acknowledgements
S. Arora would like first to thank the Council of Scientific & Industrial Research, New Delhi, Government of India (File No. 09/143(0931)/2013 EMR-I), for financial support to carry out his research work and also thank the Department of Mathematics, Indian Institute of Technology Roorkee (IIT Roorkee), for providing stimulating scientific environment and resources. M. T. Mohan would like to thank the Department of Science and Technology (DST), Govt. of India, for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). J. Dabas would like to thank the Department of Atomic Energy (DAE), Mumbai, Government of India, project (File No-02011/12/2021 NBHM(R.P)/R &D II/7995).
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Appendix A: A Variational Method
Appendix A: A Variational Method
In this section, we discuss the variational method (cf. [27,28,29,30], etc.) for finite-approximate controllability of a semilinear fractional evolution equation with finite delay. For easiness of demonstration, we are not considering impulses in the system. Let us consider the following fractional evolution equation with finite delay:
where \(^C\textrm{D}_{0,t}^{\alpha }\) denotes the Caputo fractional derivative of order \(\alpha \in (1,2)\), the operators \(\textrm{A}:\mathrm {D(A)}\subset {\mathbb {H}}\rightarrow {\mathbb {H}}\) and \(\textrm{B}:{\mathbb {U}}\rightarrow {\mathbb {H}}\) are the same as defined in (1.1). The control function \(u\in \textrm{L}^{2}(J;{\mathbb {U}})\) and the given function \(f:J\times \textrm{C}(I_c;{\mathbb {H}})\rightarrow {\mathbb {H}}\), where \(I_c=[-c,0]\). For every \(t \in J,\ x_t \in \textrm{C}(I_c;{\mathbb {H}})\) and the function \(x_t(\cdot )\) is defined by \(x_t(s)= x(t + s),\ s\in I_c\).
Definition A.1
A function \(x\in \textrm{C}(J_c;{\mathbb {H}}),\) where \(J_c=[-c,T]\), is called a mild solution of (A.1), if \(x_0=\phi \) and
Let us impose the following assumptions on the nonlinear term \(f(\cdot ,\cdot )\).
\(\textit{(A1)}\)
-
(i)
the mapping \(t\mapsto f(t, \psi ) \) is strongly measurable on J for each \( \psi \in \textrm{C}(I_c;{\mathbb {H}})\), and the function \(f(t,\cdot ): \textrm{C}(I_c;{\mathbb {H}})\rightarrow {\mathbb {H}}\) is continuous for a.e. \(t\in J\).
-
(ii)
For each positive integer r, there exists a constant \(\beta \in [0,q]\) and a function \(\varGamma _r\in \textrm{L}^{\frac{1}{\beta }}(J;\mathbb {R^{+}})\) such that
$$\begin{aligned} \sup _{\left\| \psi \right\| _{\textrm{C}(I_c;{\mathbb {H}})}\le r} \left\| f(t, \psi )\right\| _{{\mathbb {H}}}\le \varGamma _{r}(t), \text{ for } \text{ a.e. } \ t \in J \ \text{ and } \ \psi \in \textrm{C}(I_c;{\mathbb {H}}), \end{aligned}$$with
$$\begin{aligned} \liminf _{r \rightarrow \infty } \frac{\left\| \varGamma _r\right\| _{\textrm{L}^{\frac{1}{\beta }}(J;\mathbb {R^+})}}{r} = l< \infty . \end{aligned}$$
Here, we obtain the finite-approximate controllability of the semilinear system (A.1) by using the technique introduced in the works [27, 29, 30, 55], etc. For this, we consider the following functional:
for any \(\uplambda >0\), \(z\in \textrm{C}(J_c;{\mathbb {H}})\) and
where \(h\in {\mathbb {H}}\).
Lemma A.2
The set \(\textrm{V}=\{g(z):z\in {\mathcal {B}}_r\}\) is relatively compact in \({\mathbb {H}}\), where \({\mathcal {B}}_r=\{y\in \textrm{C}(J_c;{\mathbb {H}}):\left\| y\right\| _{\textrm{C}(J_c;{\mathbb {H}})}\le r\}\).
Proof
The relative compactness of the set \(\textrm{V}\) is immediately follows by the facts that the operator \(\textrm{S}_{q}(t)\) is compact for \(t\ge 0\) and also the operator \((\textrm{G}f)(\cdot ) =\int _{0}^{\cdot }(\cdot -s)^{q-1}\textrm{S}_{q}(\cdot -s)f(s)\textrm{d}s\) is compact (see Lemma 3.4, [3]). \(\square \)
Lemma A.3
The functional \({\mathcal {J}}_{\uplambda }(\cdot ,\cdot )\) satisfies the following properties.
-
(i)
The mapping \(\varphi \mapsto {\mathcal {J}}_{\uplambda }(\varphi ,z)\) is strictly convex and Gâteaux differentiable.
-
(ii)
For any \(r>0\)
$$\begin{aligned} \liminf _{\varphi \rightarrow \infty }\inf _{z\in {\mathcal {B}}_r}\frac{{\mathcal {J}}_{\uplambda }(\varphi ,z)}{\left\| \varphi \right\| _{{\mathbb {H}}}}\ge \uplambda . \end{aligned}$$(A.3)
Proof
(i) From the definition of \({\mathcal {J}}_{\uplambda }(\cdot ,\cdot )\), we see immediately that the mapping \(\varphi \mapsto {\mathcal {J}}_{\uplambda }(\varphi ,z)\) is strictly convex and Gâteaux differentiable.
(ii) In order to prove the estimate (A.3), let us consider two sequences \(\{\varphi ^m\}_{m=1}^\infty \subset {\mathbb {H}}\) and \(\{z^m\}_{m=1}^\infty \subset {\mathcal {B}}_r\) with \(\left\| \varphi ^m\right\| _{{\mathbb {H}}}\rightarrow \infty \) as \(m\rightarrow \infty \). From Lemma A.2, we know that the set \(\textrm{V}=\{g(z):z\in {\mathcal {B}}_r\}\) is relatively compact. So without loss generality, taking a subsequence \(g(z^m)\) (still denoted by \(g(z^m)\)), such that
for some \(g\in {\mathbb {H}}\). Next, we define
it is clear that \(\left\| \psi ^m\right\| _{{\mathbb {H}}}=1\). Hence, by applying the Banach–Alaoglu theorem, we can find a subsequence relabeled as \(\{\psi ^m\} _{m=1}^\infty \) such that
Using the compactness of the operator \( \textrm{S}_q(t)\) for \(t\ge 0\), we have the following convergence:
By the expression (A.2), it follows that
Case I: If either
Then, it is immediate that
Case II:If
In this case, it is again follows that
Case III: If \(\liminf \limits _{m\rightarrow \infty }\int _{0}^{T}(T-s)^{q-1}\left\| \textrm{B}^*\textrm{S}_q(T-s)^*\psi ^m\right\| ^2_{{\mathbb {U}}}\textrm{d}s=0\). Using the convergence (A.4) and the Fatou lemma, we obtain
By using the above fact, Assumption 4.4 (H5) and Theorem 3.5, we have \( {\bar{\varphi }}=0\), which implies that
Since the space \({\mathcal {D}}\) is finite-dimensional and \(\pi _{{\mathcal {D}}}\) is compact, we get that
and therefore
From the convergence (A.5) and the estimate (A.6), further we obtain
Therefor, in all the cases we get the following:
which ensures that the inequality (A.3) follows. \(\square \)
The estimate (A.3) implies that the functional \({\mathcal {J}}_{\uplambda }(\cdot ,z)\) is coercive for all \(z\in {\mathcal {B}}_r\). Thus, \({\mathcal {J}}_{\uplambda }(\cdot ,z)\) has a minimizer. By the strict convexity of \({\mathcal {J}}_{\uplambda }(\cdot ,z)\), the minimum is unique which can be found as
so that
By using the above minimum, we construct the feedback control as
In order to prove the existence of a mild solution of the system (A.1), we define a set
with the norm \(\left\| \cdot \right\| _{\textrm{C}(J;{\mathbb {H}})}\). Next, for any \(\uplambda >0\), we define an operator \({\mathcal {P}}_\uplambda :{\mathcal {E}}\rightarrow {\mathcal {E}} \) as
with the control
where
and the function \({\tilde{x}}:[-c,T]\rightarrow {\mathbb {H}}\) such that \({\tilde{x}}(t)=\phi (t), \ t\in [-c,0], \ {\tilde{x}}(t)=x(t),\ t\in J\).
In view of the definition of \({\mathcal {P}}_{\uplambda }\), it is clear that the problem of existence of a mild solution for the system (A.1) is equivalent to the operator \({\mathcal {P}}_{\uplambda }\) has a fixed point. In the next theorem, we show that the operator \({\mathcal {P}}_{\uplambda }\) has a fixed point.
Theorem A.4
If Assumptions (H1) and (A1) hold true, then for arbitrary \(\uplambda >0\), the operator \({\mathcal {P}}_{\uplambda }\) given in (A.7) has a fixed point, provided
where \(\nu =\frac{2q-\beta }{1-\beta }\) and \(\delta _{\uplambda }=\left\| \uplambda (\uplambda \textrm{I}+\Phi _{0}^{T})^{-1}\pi _{{\mathcal {D}}}\right\| _{{\mathcal {L}}({\mathbb {H}})}<1\).
Proof
Proceeding similarly as in the proof of Theorem 4.2, we can obtain that the operator \({\mathcal {P}}_{\uplambda }\) defined in (A.7) satisfies the following properties:
-
(i)
There exists \(r>0\) such that \({\mathcal {P}}_{\uplambda }(\textrm{E}_r)\subset \textrm{E}_r\), where \(\textrm{E}_r=\{x\in {\mathcal {E}}: \left\| x\right\| _{\textrm{C}(J;{\mathbb {H}})}\le r\}\).
-
(ii)
\({\mathcal {P}}_{\uplambda }\) is continuous.
-
(iii)
\({\mathcal {P}}_{\uplambda }\) is compact.
In view of these three properties and an application of Schauder’s fixed point theorem, the existence of a fixed point of the operator \({\mathcal {P}}_{\uplambda }\) follows immediately. In other words, for any \(\uplambda >0\), there exists a function \(x^\uplambda \in \textrm{C}(J_c;{\mathbb {H}}),\) such that \(x^\uplambda _0=\phi \) and
with the control
where
and the proof can be completed. \(\square \)
In order to investigate the finite-approximate controllability of the semilinear system (A.1), we assume the following:
- (A2):
-
The function \( f: J \times \textrm{C}(I_c;{\mathbb {H}}) \rightarrow {\mathbb {H}} \) satisfies Assumption (A1)(i) and there exists a function \( \varGamma \in \textrm{L}^{\frac{1}{\beta }}(J;{\mathbb {R}}^+)\) with \(\beta \in [0,q]\) such that
$$\begin{aligned} \Vert f(t,\psi )\Vert _{{\mathbb {H}}}\le \varGamma (t),\ \text { for all }\ (t,\psi ) \in J \times \textrm{C}(I_c;{\mathbb {H}}). \end{aligned}$$
Theorem A.5
If Assumptions (H1), (H5), (A2) are satisfied, then the system (A.1) is finite-approximately controllable.
Proof
Since we know that the functional \({\mathcal {J}}_{\uplambda }(\varphi ,x^\uplambda )\) has a unique minima, say \(\hat{\varphi _{\uplambda }}\in {\mathbb {H}}\) of the form
where \(g(x^\uplambda )\) given in (A.8). For any given \(\varphi \in {\mathbb {H}}\) and \(\mu \in {\mathbb {R}},\) we have
or in other words
Dividing the above inequality by \(\mu >0\) and passing \(\mu \rightarrow 0^+\), we deduce that
Repeating this argument with \(\mu <0,\) we finally obtain that
Using the above estimate, we easily find
holds for any \(\varphi \in {\mathbb {H}}\), which implies that
Next, by using Assumption (A2), we obtain for \(0<\beta \le q\)
The case \(\beta =0\) can be obtained in a similar way. The above relation ensures that the set \( \{f(\cdot , x^{\uplambda }_{(\cdot )}): \uplambda >0\}\) in \( \textrm{L}^2([0,T]; {\mathbb {H}})\) is bounded. By applying the Banach–Alaoglu theorem, we can find a subsequence \( \{f(\cdot , x^{\uplambda _i}_{(\cdot )})\}_{i=1}^{\infty }\) such that
Using the above weak convergence together with the compactness of the operator \((\textrm{G}f)(\cdot ) =\int _{0}^{\cdot }(\cdot -s)^{q-1}\textrm{S}_{q}(\cdot -s)f(s)\textrm{d}s:\textrm{L}^2(J;{\mathbb {H}})\rightarrow \textrm{C}(J;{\mathbb {H}}) \) (see Lemma 3.4 [3]), we get
where
Combining the estimates (A.9) and (A.10), we evaluate
where \(\varsigma =\min \{\langle \pi _{{\mathcal {D}}}\Phi _{0}^T\pi _{{\mathcal {D}}}\varphi ,\varphi \rangle : \left\| \pi _{{\mathcal {D}}}\varphi \right\| _{{\mathbb {H}}}=1\}>0\). Using the convergence (A.11), Assumption 4.4 (H5) and Theorem 3.5, we obtain
By taking \(\varphi \in {\mathcal {D}}\) in the estimate (A.10), we get
Thus, the system (A.1) is finite-approximately controllable. \(\square \)
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Arora, S., Mohan, M.T. & Dabas, J. Finite-Approximate Controllability of Impulsive Fractional Functional Evolution Equations of Order \(1<\alpha <2\). J Optim Theory Appl 197, 855–890 (2023). https://doi.org/10.1007/s10957-023-02205-4
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DOI: https://doi.org/10.1007/s10957-023-02205-4
Keywords
- Finite-approximate controllability
- Cosine family
- Non-instantaneous impulses
- Caputo fractional derivative
- Mainardi’s Wright-type function