Partial stability analysis of stochastic differential equations with a general decay rate

Abstract

This paper is concerned with the almost sure partial practical stability of stochastic differential equations with general decay rate. We establish some sufficient conditions based upon the construction of appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate the efficiency of the obtained results.

Appendices

Appendix

Some details of calculus used in the proof of Theorem 3.3 are provided in what follows:

Appendix A: Details of calculus of Eq. (3.3)

$$\begin{aligned} \lambda (t)^m ||{x_1}(t)||^q-\rho (t)&=\lambda (t)^m\Big (||{x_1}(t)||^q -\Big (\Big (\dfrac{\rho (t)}{\lambda (t)^m}\Big )^{\frac{1}{q}}\Big )^q\Big )\\&=\lambda (t)^m\Big (||{x_1}(t)||-\Big (\dfrac{\rho (t)}{\lambda (t)^m}\Big )^\frac{1}{q}\Big ) \Big (||{x_1}(t)||^{q-1}+||{x_1}(t)||^{q-2}\Big (\dfrac{\rho (t)}{\lambda (t)^m} \Big )^{\frac{1}{q}}\\&\quad +\cdots +\Big (\dfrac{\rho (t)}{\lambda (t)^m}\Big )^{\frac{q-1}{q}}\Big )\\&=\lambda (t)^m\Big (||{x_1}(t)||-\Big (\dfrac{\rho (t)}{\lambda (t)^m} \Big )^{\frac{1}{q}}\Big )\sum _{k=1}^{q}||{x_1}(t)||^{q-k} \Big (\dfrac{\rho (t)}{\lambda (t)^m}\Big )^{\frac{k-1}{q}}. \end{aligned}$$

Appendix B: Details of calculus of Eq. (3.5)

$$\begin{aligned} \ln ({{\mathcal {V}}}(t,x(t)))\le & {} \ln ({{\mathcal {V}}}(T,x(T))) +\int _T^t \dfrac{\psi _1 {{\mathcal {V}}}(s,x(s))+\rho (s)}{{{\mathcal {V}}}(s,x(s))}\mathrm{{d}}s +{{\mathcal {M}}}(t)-\frac{1}{2}\int _T^t\dfrac{{\mathcal {H}}{{\mathcal {V}}}(s,x(s))}{{{\mathcal {V}}}^2(s,x(s))}\mathrm{{d}}s\\\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\int _T^t \psi _1(s)\mathrm{{d}}s+ \int _T^t \dfrac{\rho (s)}{{{\mathcal {V}}}(s,x(s))}\mathrm{{d}}s+{{\mathcal {M}}}(t) -\frac{1}{2}\int _T^t\dfrac{{\mathcal {H}}{{\mathcal {V}}}(s,x(s))}{{{\mathcal {V}}}^2(s,x(s))}\mathrm{{d}}s\\\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\int _T^t\psi _1(s)\mathrm{{d}}s+ \int _T^t \dfrac{\rho (s)}{\lambda (s)^m ||{x_1}(s)||^q}\mathrm{{d}}s +{{\mathcal {M}}}(t) -\frac{1}{2}\int _T^t\dfrac{{\mathcal {H}}{{\mathcal {V}}}(s,x(s))}{{{\mathcal {V}}}^2(s,x(s))}\mathrm{{d}}s. \end{aligned}$$

Appendix C: Details of calculus of Eq. (3.7)

$$\begin{aligned} \ln ({{\mathcal {V}}}(t,x(t)))\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\int _T^t\psi _1(s)\mathrm{{d}}s + (t-T)+\frac{2}{\beta }\ln (k-1) +\frac{\beta }{2}\int _{T}^{t}\frac{{\mathcal {H}} {{\mathcal {V}}}(s,x(s))}{{{\mathcal {V}}}^{2}(s,x(s))}\mathrm{{d}}s\\&\quad -\frac{1}{2}\int _{T}^{t}\frac{{\mathcal {H}}{{\mathcal {V}}} (s,x(s))}{{{\mathcal {V}}}^{2} (s,x(s))}\mathrm{{d}}s\\\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\int _T^t\psi _1(s)\mathrm{{d}}s+(t-T)+\frac{2}{\beta }\ln (k-1)-\frac{1-\beta }{2}\int _{T}^{t}\frac{{\mathcal {H}}{{\mathcal {V}}}(s,x(s))}{{{\mathcal {V}}}^{2}(s,x(s))}\mathrm{{d}}s\\\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\int _T^t\psi _1(s)\mathrm{{d}}s+(t-T)+\frac{2}{\beta } \ln (k-1) -\frac{1-\beta }{2}\int _{T}^{t} \psi _2(s) \mathrm{{d}}s\nonumber \\&\quad -\frac{1-\beta }{2}\int _{T}^{t} \frac{\xi }{{{\mathcal {V}}}^{2}(s,x(s))}\mathrm{{d}}s\\\le & {} \ln ({{\mathcal {V}}}(T,x(T)))+\frac{2}{\beta }\ln (k-1)+\int _T^t\psi _1(s)\mathrm{{d}}s +(t-T) -\frac{1-\beta }{2}\int _{T}^{t} \psi _2(s) \mathrm{{d}}s. \end{aligned}$$