{"id":18881,"date":"2023-11-11T11:11:34","date_gmt":"2023-11-11T10:11:34","guid":{"rendered":"https:\/\/djalil.chafai.net\/blog\/?p=18881"},"modified":"2024-01-14T16:42:08","modified_gmt":"2024-01-14T15:42:08","slug":"back-to-basics-odes-lyapounov-stability","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2023\/11\/11\/back-to-basics-odes-lyapounov-stability\/","title":{"rendered":"Back to basics : ODEs Lyapunov stability"},"content":{"rendered":"<figure id=\"attachment_18885\" aria-describedby=\"caption-attachment-18885\" style=\"width: 374px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.annales.org\/archives\/x\/jordan.html\"><img loading=\"lazy\" class=\"wp-image-18885 size-full\" src=\"http:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2024\/01\/JordanX.jpg\" alt=\"Caricature of Camille Jordan (1838 - 1922) by one of his students at \u00c9cole Polytechnique, during a lecture on differential calculus : \u00ab J'ai mis \u2202u mais c'est une faute d'impression \u00bb.\" width=\"374\" height=\"413\" srcset=\"https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2024\/01\/JordanX.jpg 374w, https:\/\/djalil.chafai.net\/blog\/wp-content\/uploads\/2024\/01\/JordanX-272x300.jpg 272w\" sizes=\"(max-width: 374px) 100vw, 374px\" \/><\/a><figcaption id=\"caption-attachment-18885\" class=\"wp-caption-text\">Caricature of Camille Jordan (1838 - 1922) by one of his students at \u00c9cole Polytechnique, during a lecture on differential calculus : \u00ab J'ai mis \u2202u mais c'est une faute d'impression \u00bb.<\/figcaption><\/figure>\n<p style=\"text-align:justify;\">This tiny post is about stability in the sense of Lyapunov of stationary points of autonomous ordinary differential equation (ODE) in $\\mathbb{R}^n$ given by \\[ x'(t)=f(x(t)),\\quad x(0)=x_0\\] where $f:O\\subset\\mathbb{R}^n\\to\\mathbb{R}^n$ is a $\\mathcal{C}^1$ vector field. This includes two important examples:<\/p>\n<ul>\n<li><strong>Gradient flows<\/strong> : $f=-\\nabla\\mathcal{E}$ where $\\mathcal{E}:O\\to\\mathbb{R}$ is $\\mathcal{C}^2$<br \/>\nGeometrically, the system descents down the levelsets of $\\mathcal{E}$<\/li>\n<li><strong>Hamiltonian systems<\/strong> : $f=\\Omega\\nabla H$ with $\\Omega$ skew-symmetric and $H:O\\to\\mathbb{R}$ is $\\mathcal{C}^2$<br \/>\nGeometrically, the system remains in the levelset of $H$ where it started\n<\/li>\n<\/ul>\n<p style=\"text-align:justify;\">The $\\mathcal{C}^1$ regularity of $f$ ensures, via the Cauchy-Lipschitz or Picard\u2013Lindel\u00f6f theorem, the local existence and uniqueness of solutions for all $x_0\\in O$. As a consequence, the solution that takes the value $x$ at initial time $0$, denoted $t\\mapsto\\varphi_t(x)$, is uniquely defined on a maximal time interval ending up at $T_\\max(x)$, and blows as $t\\to T_\\max(x)$ if $T_\\max(x)<\\infty$. We say that $\\varphi_t:O\\to O$ is the flow of the ODE. The phase space $O$ is partitioned into disjoint integral curves associated to the solutions. We are interested in the long time behavior of solutions near stationary points.<\/p>\n<p style=\"text-align:justify;\"><strong>Stability of stationary points.<\/strong> Suppose that $x_0$ is stationary : $f(x_0)=0$. Then $x(t)=x_0$ for all $t$. We say that this stationary point is <strong>stable<\/strong> in the sense of Lyapunov when for all $\\varepsilon>0$, there exists $\\eta>0$ such that for all $x\\in\\overline{B}(x_0,\\eta)$,<br \/>\n\\[<br \/>\nT_\\max(x)=+\\infty<br \/>\n\\quad\\text{and}\\quad<br \/>\n\\|\\varphi_t(x)-x_0\\|\\leq\\varepsilon\\text{ for all $t\\geq0$},<br \/>\n\\] the solution remains as close to $x_0$ as we want and for all time provided that we start close enough. We say that the stationary point $x_0$ is unstable otherwise. Moreover, we say that a stable stationary point $x_0$ is <strong>asymptotically stable<\/strong> if for some $\\eta_0>0$ and all $x\\in\\overline{B}(x_0,\\eta_0)$,<br \/>\n\\[<br \/>\n\\lim_{t\\to\\infty}\\varphi_t(x)=x_0.<br \/>\n\\] Furthermore, we say that it is <strong>exponentially stable<\/strong> if there exist positive constants $\\alpha,c,C>0$ such that forall $x\\in\\overline{B}(x_0,\\eta_0)$ and all $t\\geq0$,<br \/>\n\\[<br \/>\n\\|\\varphi_t(x)-x_0\\|\\leq C\\mathrm{e}^{-\\alpha t}\\|x-x_0\\|.<br \/>\n\\]<\/p>\n<p style=\"text-align:justify;\"><strong>Lyapunov function.<\/strong> It is a $\\mathcal{C}^2$ function $\\Phi:O\\to\\mathbb{R}$ such that for all $x\\in O$,<br \/>\n\\[<br \/>\n\\langle\\nabla \\Phi(x),f(x)\\rangle\\leq0.<br \/>\n\\] This is equivalent to say that $t\\mapsto\\Phi(\\varphi_t(x))$ is non-increasing for all $x$, since $\\varphi_0(x)=x$ and<br \/>\n\\[<br \/>\n\\partial_t\\Phi(\\varphi_t(x))=\\langle\\nabla\\Phi(\\varphi_t(x),\\varphi_t'(x)\\rangle<br \/>\n=\\langle\\nabla\\Phi(\\varphi_t(x)),f(\\varphi_t(x))\\rangle\\leq0.<br \/>\n\\] For a gradient flow $f=-\\nabla\\Phi$, the function $\\Phi$ is a Lyapunov function and in this case<br \/>\n\\[<br \/>\n\\partial_t\\Phi(\\varphi_t(x))=-\\|\\nabla\\Phi(\\varphi_t(x))\\|^2.<br \/>\n\\]<\/p>\n<p style=\"text-align:justify;\"><strong>Prime integral.<\/strong> It is a $\\mathcal{C}^2$ function $H:O\\to\\mathbb{R}$ such that for all $x\\in O$,<br \/>\n\\[<br \/>\n\\langle\\nabla H(x),f(x)\\rangle=0.<br \/>\n\\] This is equivalent to say that both $H$ and $-H$ are Lyapunov functions, and also to say that $t\\mapsto H(\\varphi_t(x))$ is constant for all $t$ and $x$. For a Hamiltonian system $f=\\Omega\\nabla H$, the Hamiltonian $H$ is a prime integral, indeed thanks to the skew-symmetry of $\\Omega$,<br \/>\n\\[<br \/>\n\\partial_t H(\\varphi_t(x))<br \/>\n=\\langle\\nabla H(\\varphi_t(x)),f(\\varphi_t(x))\\rangle<br \/>\n=\\langle\\nabla H(\\varphi_t(x),\\Omega\\nabla H(\\varphi_t(x))\\rangle<br \/>\n=0.<br \/>\n\\]<\/p>\n<p style=\"text-align:justify;\"><strong>Criterion for stability.<\/strong>. If $x_0$ is a strict local minimum of a Lyapunov function $\\Phi$ then $x_0$ is stationary and stable. Actually the proof shows that we only need to know that $\\Phi$ is locally a Lyapunov function, namely Lyapunov on a small enough ball centered at $x_0$ included in $O$, this modification of the open set on which we define the ODE has potentially an impact on $T_\\max$, but at the end, we conclude that $T_\\max=+\\infty$ and it does not matter.<\/p>\n<p style=\"text-align:justify;\"><strong>Proof.<\/strong> We use a reasoning by continuity. Since $x_0$ is a strict minimum, for $\\varepsilon>0$ small enough, $\\overline{B}(x_0,\\varepsilon)\\subset O$ and $\\Phi(x_0)<\\Phi(x)$ for all $x\\in\\overline{B}(x_0,\\varepsilon)\\setminus\\{x_0\\}$. Since the sphere $\\overline{S}(x_0,\\varepsilon)$ is compact because the dimension is finie, and $\\Phi$ is continuous, there exists $x_1\\in S(x_0,\\varepsilon)$ such that $m:=\\min_{S(x_0,\\varepsilon)}\\Phi=\\Phi(x_1)$, and since $\\Phi(x_1)>\\Phi(x_0)$, we get $m>\\Phi(x_0)$. On the other hand, since $\\Phi$ is continuous at $x_0$ and $\\Phi(x_0)< m$, there exists $0<\\eta<\\varepsilon$ such that the maximum $M$ of $\\Phi$ on $\\overline{B}(x_0,\\eta)$ is $< m$. Now, for all $x\\in\\overline{B}(x_0,\\eta)$, since $\\varphi_0(x)=x$ and $t\\mapsto\\Phi(\\varphi_t(x))$ is non-increasing because $\\Phi$ is a Lyapunov function, the function $\\Phi$ remains $\\leq M< m$ along the trajectory, hence the trajectory cannot reach the sphere $S(x_0,\\varepsilon)$, thus it remains in $B(x_0,\\varepsilon)$. Therefore $x_0$ is stable. We have proved stability before stationarity. To show that it is stationary, it suffices to linearize the flow as $\\varphi_t(x)-x_0=tf(x)+o_{t\\to0}(t)$ and to take $\\eta=t\\eta'$. Finally, since the trajectory is bounded and the dimension is finite, we have necessarily $T_{\\max}(x)=+\\infty$.<\/p>\n<p style=\"text-align:justify;\"><strong>Criterion for being exponentially stable.<\/strong>. If $x_0$ is stationary and<br \/>\n\\[<br \/>\n\\max\\Re\\mathrm{spec}(\\mathrm{Jac}(f)(x_0))&lt;0,<br \/>\n\\]then $x_0$ is exponentially stable. This is a spectral criterion by linearization of the ODE.<\/p>\n<p style=\"text-align:justify;\"><strong>Proof.<\/strong> Let $A:=\\mathrm{Jac}(f)(x_0)$ be the Jacobian matrix of $f$ at $x_0$, and its Jordan reduction<br \/>\n\\[<br \/>\nA=P^{-1}(D+E)P,<br \/>\n\\quad D:=\\mathrm{Diag}(\\lambda_1,\\ldots,\\lambda_n),\\quad<br \/>\nP\\in\\mathrm{GL}_n(\\mathbb{C}),\\quad E\\in\\mathcal{M}_n(\\mathbb{R}).<br \/>\n\\] The vector $x_0$ (and $x$ below) and the matrix $A$ are real but $P$ and $D$ are complex ($E$ is real and $DE=ED$ but we do not use that). We can always assume that $\\|E\\|$ is arbitrary small since the Jordan blocs can be conjugated by diagonal matrices, at the price of making $\\|P\\|$ bigger:<br \/>\n\\[<br \/>\n\\begin{bmatrix}<br \/>\n\\lambda &amp; 1 &amp; 0 &amp; \\cdots &amp; 0 \\\\<br \/>\n0 &amp; \\lambda &amp; 1 &amp; \\cdots &amp; 0 \\\\<br \/>\n\\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\<br \/>\n0 &amp; 0 &amp; 0 &amp; \\lambda &amp; 1 \\\\<br \/>\n0 &amp; 0 &amp; 0 &amp; 0 &amp; \\lambda<br \/>\n\\end{bmatrix}<br \/>\n=<br \/>\n\\mathrm{Diag}(1,\\varepsilon,\\varepsilon^2,\\ldots)<br \/>\n\\begin{bmatrix}<br \/>\n\\lambda &amp; \\varepsilon &amp; 0 &amp; \\cdots &amp; 0 \\\\<br \/>\n0 &amp; \\lambda &amp; \\varepsilon &amp; \\cdots &amp; 0 \\\\<br \/>\n\\vdots &amp; \\vdots &amp; \\vdots &amp; \\ddots &amp; \\vdots \\\\<br \/>\n0 &amp; 0 &amp; 0 &amp; \\lambda &amp; \\varepsilon \\\\<br \/>\n0 &amp; 0 &amp; 0 &amp; 0 &amp; \\lambda<br \/>\n\\end{bmatrix}<br \/>\n\\mathrm{Diag}(1,\\varepsilon^{-1},\\varepsilon^{-2},\\ldots).<br \/>\n\\] We have $\\alpha:=\\max\\Re\\mathrm{spec}(\\mathrm{Jac}(f)(x_0))=\\max_{1\\leq k\\leq<br \/>\nn}\\Re\\lambda_k&lt;0$. Now<br \/>\n\\[<br \/>\n\\Phi(x):=\\|P(x-x_0)\\|^2=\\langle \\overline{P}^\\top P(x-x_0),x-x_0\\rangle,<br \/>\n\\] hence, using the fact that $x-x_0$ is real for the first equality,<br \/>\n\\[<br \/>\n\\nabla\\Phi(x)=2\\Re(\\overline{P}^\\top P)(x-x_0)<br \/>\n\\quad\\text{and in particular}\\quad<br \/>\n\\left\\|\\nabla\\Phi(x)\\right\\|=O(\\|x-x_0\\|).<br \/>\n\\] On the other hand, $f(x_0)=0$ since $x_0$ is stationary and thus<br \/>\n\\[<br \/>\nf(x)=A(x-x_0)+o(\\|x-x_0\\|).<br \/>\n\\] We have then, using the fact that $A$ and $x-x_0$ are real for the first equality,<br \/>\n\\begin{align*}<br \/>\n\\langle\\nabla\\Phi(x),f(x)\\rangle<br \/>\n&amp;=2\\Re\\langle\\overline{P}^\\top P(x-x_0),A(x-x_0)\\rangle+o(\\|x-x_0\\|^2)\\\\<br \/>\n&amp;=2\\Re\\langle P(x-x_0),(D+E)P(x-x_0)\\rangle+o(\\|x-x_0\\|^2)\\\\<br \/>\n&amp;=2\\Re\\langle z,Dz\\rangle+2\\Re\\langle z,Ez\\rangle+o(\\|x-x_0\\|^2)<br \/>\n\\end{align*} where $z:=P(x-x_0)$. But now by the Cauchy--Schwarz inequality<br \/>\n\\[<br \/>\n2\\Re\\langle z,Ez\\rangle\\leq2\\|E\\|\\|z\\|^2,<br \/>\n\\] while<br \/>\n\\[<br \/>\n\\langle z,Dz\\rangle =\\langle z,(\\Re<br \/>\nD)z\\rangle-\\mathrm{i}\\langle z,(\\Im D)z\\rangle =\\sum_k|z_k|^2\\Re<br \/>\nD_{k,k}-\\mathrm{i}\\sum_k|z_k|^2\\Im D_{k,k}<br \/>\n\\] gives<br \/>\n\\[<br \/>\n2\\Re\\langle z,Dz\\rangle\\leq-2\\alpha\\|z\\|^2.<br \/>\n\\] Thus, for $\\beta&lt;2\\alpha$, $\\|E\\|$ small enough, and $x\\in\\overline{B}(x_0,\\eta_0)$, with $\\eta_0>0$ small enough, we get<br \/>\n\\begin{align*}<br \/>\n\\langle\\nabla\\Phi(x),f(x)\\rangle<br \/>\n&amp;\\leq-2\\alpha\\|P(x-x_0)\\|^2+2\\|E\\|\\|P(x-x_0)\\|^2+o(\\|x-x_0\\|^2)\\\\<br \/>\n&amp;\\leq-\\beta\\|P(x-x_0)\\|^2=-\\beta\\Phi(x).<br \/>\n\\end{align*} In particular $\\Phi$ is a Lyapunov function, on the restricted set $O_0:=B(x_0,\\eta_0)\\subset O$, with a strict minimum at $x_0$ because $P$ is invertible, hence $x_0$ is stable. Moreover $\\partial_t\\Phi(\\varphi_t(x))\\leq-\\beta\\Phi(\\varphi_t(x))$ for $x\\in\\overline{B}(x_0,\\eta_0)$, and the Gr\u00f6nwall lemma below gives<br \/>\n\\[<br \/>\n\\Phi(\\varphi_t(x))\\leq\\Phi(\\varphi_0(x))\\mathrm{e}^{-\\beta t}=\\Phi(x)\\mathrm{e}^{-\\beta t}.<br \/>\n\\] Next, since $P$ is invertible, there exists a positive constant $C>0$ such that<br \/>\n\\[<br \/>\nC^{-1}\\|x-x_0\\|^2\\leq\\Phi(x)\\leq C\\|x-x_0\\|^2,<br \/>\n\\] hence for $x\\in\\overline{B}(x_0,\\eta_0)$ and all $t\\geq0$,<br \/>\n\\[<br \/>\n\\|\\varphi_t(x)-x_0\\|\\leq C^2\\mathrm{e}^{-\\tfrac{\\beta}{2} t}\\|x-x_0\\|<br \/>\n\\] which means that $x_0$ is exponentially stable.\n<\/p>\n<p style=\"text-align:justify;\"><strong>Gr\u00f6nwall lemma.<\/strong>. If $u:[a,b]\\to\\mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if<br \/>\n\\[<br \/>\nu'(t)\\leq\\alpha(t)u(t)<br \/>\n\\] for all $t\\in(a,b)$, with $\\alpha:[a,b]\\to\\mathbb{R}$ continuous, then for all $t\\in[a,b]$,<br \/>\n\\[<br \/>\nu(t)\\leq u(a)\\exp\\bigr(\\int_a^t\\alpha(s)\\mathrm{d}s\\bigr).<br \/>\n\\] The equality case $u'(t)=\\alpha(t)u(t)$ is a one-dimensional non-autonomous linear ODE. To see it, we observe that the solution of the equality case is positive : $v(t):=\\mathrm{e}^{\\int_a^t\\alpha(s)\\mathrm{d}s}>0$ for all $t\\in[a,b]$, and $v'(t)=\\alpha(t)v(t)$ for all  $t\\in(a,b)$, while $v(a)=1$, hence<br \/>\n\\[<br \/>\n\\Bigr(\\frac{u(t)}{v(t)}\\Bigr)'<br \/>\n=\\frac{u'(t)v(t)-u(t)v'(t)}{v(t)^2}<br \/>\n=\\frac{(u'(t)-\\alpha(t)u(t))v(t)}{v(t)^2}<br \/>\n\\leq0<br \/>\n\\] for all $t\\in(a,b)$, therefore $u(t)\/v(t)\\leq u(a)\/v(a)=u(a)$ for all $t\\in[a,b]$.<\/p>\n<p style=\"text-align:justify;\">It seems that it was already known to Alexandre Lyapunov (1857 - 1918) as well as to Henri Poincar\u00e9 (1854 - 1912) that the solution of a linear non-autonomous ODE $x'(t)=A(t)x(t)$ in dimension $n\\geq2$ can perfectly diverge as $t\\to\\infty$ while $A(t)$ has all eigenvalues $&lt;0$ for all $t$.<\/p>\n<p style=\"text-align:justify;\"><strong>Note.<\/strong> This tiny post is taken from a course on topology and differential calculus, that I have the pleasure of teaching, succeeding to my former colleague Dmitry Chelkak.<\/p>\n<p style=\"text-align:justify;\"><strong>Further reading.<\/strong><\/p>\n<ul>\n<li>\nAlexandre Liapounoff<br \/>\n<a href=\"http:\/\/www.numdam.org\/item\/AFST_1907_2_9__203_0\/\">Probl\u00e8me g\u00e9n\u00e9ral de la stabilit\u00e9 du mouvement<\/a><br \/>\nAnnales de la Facult\u00e9 des sciences de Toulouse : Math\u00e9matiques 9 (1907) 203-474<br \/>\nTraduit en fran\u00e7ais, du russe, par \u00c9douard Davaux, ing\u00e9nieur de la Marine \u00e0 Toulon. \u00ab <em>M. Liapounoff a tr\u00e8s gracieusement autoris\u00e9 la publication en langue fran\u00e7aise de son M\u00e9moire imprim\u00e9 en 1892 par la Soci\u00e9t\u00e9 math\u00e9matique de Kharkow. La traduction a \u00e9t\u00e9 revue et corrig\u00e9e par l\u2019auteur qui y a ajout\u00e9 une Note r\u00e9dig\u00e9e d\u2019apr\u00e8s un Article paru en 1893 dans les Communications de la Soci\u00e9t\u00e9 math\u00e9matique de Kharkow.<\/em>\u00bb\n<\/li>\n<li>\nCamille Jordan<br \/>\n<a href=\"https:\/\/gallica.bnf.fr\/ark:\/12148\/bpt6k65441537\">Trait\u00e9 des substitutions et des \u00e9quations alg\u00e9briques<\/a><br \/>\nGauthier-Villars (1870)\n<\/li>\n<li>\nVladimir I. Arnold<br \/>\n<a href=\"https:\/\/en.wikipedia.org\/wiki\/Mathematical_Methods_of_Classical_Mechanics\">Mathematical methods of classical mechanics<\/a><br \/>\nSpringer (1978)\n<\/li>\n<li>\nTwo other related posts on this blog<br \/>\n<a href=\"https:\/\/djalil.chafai.net\/blog\/2023\/10\/10\/back-to-basics-hamiltonian-systems\/\">Back to basics : Hamiltonian systems<\/a><br \/>\n<a href=\"https:\/\/djalil.chafai.net\/blog\/2023\/07\/19\/unstable-solutions-of-nonautonomous-linear-odes\/\">Unstable solutions of Nonautonomous Linear ODEs<\/a>\n<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>This tiny post is about stability in the sense of Lyapunov of stationary points of autonomous ordinary differential equation (ODE) in $\\mathbb{R}^n$ given by \\[&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2023\/11\/11\/back-to-basics-odes-lyapounov-stability\/\">Continue reading<span class=\"screen-reader-text\">Back to basics : ODEs Lyapunov stability<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":529},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/18881"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=18881"}],"version-history":[{"count":109,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/18881\/revisions"}],"predecessor-version":[{"id":19084,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/18881\/revisions\/19084"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=18881"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=18881"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=18881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}