3 Systems
The most general systems the library considers are \(\dot x = F(t, x, u)\) in continuous time and \(x^{+} = F(k, x, u)\) in discrete time, each with an output \(y = h(t, x, u)\). As for the linear systems of the next chapter, there is one object per time domain: the data has the same shape, but a continuous trajectory is an integral curve and a discrete one is a recursion.
Special cases are constructors on the general object, each with a predicate recognising it: time invariance drops \(t\), and autonomy drops \(u\). Autonomy is used here in the control-theory sense, without input; much of the dynamical-systems literature uses the word for time invariance instead.
3.1 Continuous time
A continuous-time system with state space \(E\), input space \(U\) and output space \(Y\) is the data of two maps
to be read as \(\dot x = F(t, x, u)\), \(y = h(t, x, u)\). The structure is only the data.
The continuous-time system \(\dot x = f(x, u)\), \(y = g(x, u)\), whose equations do not depend on time.
A continuous-time system is time-invariant when \(F(t, \cdot , \cdot )\) and \(h(t, \cdot , \cdot )\) are the same for every \(t\).
The system of definition 66 is time-invariant (definition 67).
A time-invariant continuous-time system (definition 67) is \(\operatorname {timeInvariant} f\, g\) for some \(f\) and \(g\). The converse of lemma 68.
Read \(f\) and \(g\) off at \(t = 0\); time invariance says every other time agrees with it.
The continuous-time system \(\dot x = f(t, x)\), \(y = g(t, x)\), which has no input: its input space is the one-point type.
A continuous-time system is autonomous when \(F(t, x, u)\) and \(h(t, x, u)\) do not depend on \(u\).
The system of definition 70 is autonomous (definition 71).
If a continuous-time system is autonomous (definition 71) and its input space is nonempty, there are \(f\) and \(g\) with \(F(t, x, u) = f(t, x)\) and \(h(t, x, u) = g(t, x)\) for every input \(u\).
Fix any input \(u_0\) and read \(f\) and \(g\) off at it.
The autonomous system (definition 70) \(\dot x = F(t, x, 0)\), \(y = h(t, x, 0)\) obtained by holding the input at \(0\). Stability of a system with input means stability of this system.
A map is a trajectory of the unforced system (definition 74) on \(I\) exactly when it is a trajectory of \(s\) under the zero input on \(I\).
Well-posedness is kept apart from stability. The stability notions quantify over every solution, so they need no uniqueness; forward completeness enters only where a limit as \(t \to \infty \) is taken. Theorems that need either property assume it.
A continuous-time system is forward complete from a set \(S\) at time \(t_0\) under the input \(u\) when every trajectory on a segment \([t_0, t_1]\) (definition 83) with \(x(t_0) \in S\) agrees on that segment with a trajectory on \([t_0, \infty )\).
A continuous-time system is forward complete under the input \(u\) when it is forward complete (definition 76) from every state at every initial time.
A continuous-time system has unique solutions under the input \(u\) when any two trajectories on a segment \([t_0, t_1]\) (definition 83) that agree at \(t_0\) agree on all of \([t_0, t_1]\).
If \(x \mapsto F(t, x, u(t))\) is \(K\)-Lipschitz for every \(t\), the system has unique solutions under \(u\) (definition 78).
Grönwall’s inequality, in the form of Mathlib’s ODE_solution_unique_of_mem_Icc_right: a derivative within \([t_0, t_1]\) at a point of \([t_0, t_1)\) is a right derivative there.
If \((t, x) \mapsto F(t, x, u(t))\) is continuous and \(K\)-Lipschitz in \(x\) uniformly in \(t\), the system is forward complete under \(u\) (definition 77).
From the endpoint \(x(t_0)\) of a segment, Picard–Lindelöf gives a solution on \([t_0, \infty )\) (theorem 46), which agrees with the segment by uniqueness (theorem 79).
Under an input signal \(u : \mathbb {R} \to U\), the right-hand side of the state equation is the time-dependent vector field \((t, x) \mapsto F(t, x, u(t))\).
Under an input signal \(u\), the output equation is \((t, x) \mapsto h(t, x, u(t))\). An instantaneous relation: no derivative appears.
A map \(x : \mathbb {R} \to E\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {R}\) when
the derivative being taken within \(I\). Stated on a set: a map can satisfy the equation on an interval without being defined for all time.
A trajectory (definition 83) defined on all of \(\mathbb {R}\).
definition 83 at \(I = \mathbb {R}\) is definition 84.
A trajectory for all time is a trajectory on every set.
A set is forward invariant when trajectories that start in it stay in it.
A set \(S \subseteq E\) is forward invariant (or positively invariant) for \(s\) under the input \(u\) when every trajectory on a segment \([t_0, t_1]\) (definition 83) that starts in \(S\) stays in \(S\):
The derivative of a function along the system is the rate at which it changes along trajectories. It takes the input value as an argument: stability uses the input-free system, input-to-state notions a general input.
For \(V : \mathbb {R} \times E \to \mathbb {R}\) and an input value \(u \in U\), the derivative of \(V\) along the system \(\dot x = F(t, x, u)\) is
the Lie derivative of \(V\) along the field \(F(\cdot , \cdot , u)\) (definition 52).
The derivative form above excludes the solutions produced by discontinuous inputs, which have corners where the input jumps. The integral form admits them, and is the one input-driven notions such as input-to-state stability are stated with; for continuous right-hand sides the two agree.
A map \(x : \mathbb {R} \to E\), with \(E\) a Banach space, is a Carathéodory trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {R}\) when, for all \(a \le b\) in \(I\), the map \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) is integrable on \([a, b]\) and
Unlike definition 83 it does not require \(x\) to be differentiable, so it admits the solutions produced by discontinuous or merely measurable inputs, which input-to-state stability quantifies over.
A Carathéodory trajectory (definition 89) on all of \(\mathbb {R}\).
A Carathéodory trajectory on \(I\) is one on every subset of \(I\); in particular one for all time is one on every set.
Let \(t_0 \le t_1\) and let \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) be continuous on \([t_0, t_1]\). Then \(x\) is a Carathéodory trajectory on \([t_0, t_1]\) (definition 89) if and only if it is a trajectory there (definition 83).
Both are equivalent to the integral equation anchored at \(t_0\) (definition 31), by lemma 36. The anchored equation gives the equation between any \(a \le b\) by subtracting its values at \(b\) and \(a\).
Let \(I \subseteq \mathbb {R}\) be an interval (order-connected) and let \(\tau \mapsto F(\tau , x(\tau ), u(\tau ))\) be continuous on \(I\). Then \(x\) is a Carathéodory trajectory on \(I\) (definition 89) if and only if it is a trajectory there (definition 83).
Order-connectedness cannot be dropped: on \(\{ 0\} \cup [1, 2]\) the integral form relates \(x(1)\) to \(x(0)\), the derivative form does not.
Both notions reduce to closed segments inside \(I\), where they agree by theorem 92. For the integral form, the equation between \(a \le b\) only involves \([a, b] \subseteq I\). For the derivative form, at \(t \in I\) choose \(a \le t \le b\) in \(I\), taking \(a {\lt} t\) if \(I\) has a point left of \(t\) and \(a = t\) otherwise, and likewise for \(b\). Then \([a, b] \subseteq I\) agrees with \(I\) near \(t\), so a derivative within \([a, b]\) at \(t\) is one within \(I\).
3.2 Discrete time
A discrete-time system with state space \(E\), input space \(U\) and output space \(Y\) is the data of two maps
to be read as \(x^{+} = F(k, x, u)\), \(y = h(k, x, u)\). The same data as definition 65 with time in \(\mathbb {N}\); what differs is the reading, a shift rather than a derivative.
The discrete-time system \(x^{+} = f(x, u)\), \(y = g(x, u)\), whose equations do not depend on time.
A discrete-time system is time-invariant when \(F(k, \cdot , \cdot )\) and \(h(k, \cdot , \cdot )\) are the same for every \(k\).
The system of definition 95 is time-invariant (definition 96).
A time-invariant discrete-time system (definition 96) is \(\operatorname {timeInvariant} f\, g\) for some \(f\) and \(g\). The converse of lemma 97.
Read \(f\) and \(g\) off at \(k = 0\); time invariance says every other time agrees with it.
The discrete-time system \(x^{+} = f(k, x)\), \(y = g(k, x)\), which has no input: its input space is the one-point type.
A discrete-time system is autonomous when \(F(k, x, u)\) and \(h(k, x, u)\) do not depend on \(u\).
The system of definition 99 is autonomous (definition 100).
If a discrete-time system is autonomous (definition 100) and its input space is nonempty, there are \(f\) and \(g\) with \(F(k, x, u) = f(k, x)\) and \(h(k, x, u) = g(k, x)\) for every input \(u\).
Fix any input \(u_0\) and read \(f\) and \(g\) off at it.
Under an input sequence \(u : \mathbb {N} \to U\), the right-hand side of the state equation is the time-dependent next-state map \((k, x) \mapsto F(k, x, u(k))\). The same expression as definition 81; what differs is what a trajectory does with it, a shift rather than a derivative.
Under an input sequence \(u\), the output equation is \((k, x) \mapsto h(k, x, u(k))\).
A sequence \(x : \mathbb {N} \to E\) is a trajectory of \(s\) under the input \(u\) on a set \(I \subseteq \mathbb {N}\) when
Membership constrains the step out of \(k\), so \(I = [k_0, \infty )\) is the forward recursion from \(k_0\), and the horizon \([k_0, k_1]\) corresponds to \(I = [k_0, k_1)\).
A trajectory (definition 105) satisfying the recursion at every \(k \in \mathbb {N}\).
definition 105 at \(I = \mathbb {N}\) is definition 106.
A trajectory at every time is a trajectory on every set.
Unlike continuous time, a trajectory needs no hypothesis to exist or to be unique: the recursion defines it from any initial time and state for all later times. So no discrete-time solution escapes in finite time, and nothing like forward completeness is ever needed.
For every input \(u\), initial time \(k_0\) and state \(x_0\) there is a trajectory (definition 105) on \([k_0, \infty )\) with \(x(k_0) = x_0\).
Define it by the recursion itself.
Two trajectories (definition 105) on \([k_0, \infty )\) under the same input with \(x(k_0) = y(k_0)\) satisfy \(x(k) = y(k)\) for every \(k \ge k_0\).
Induction on \(k\).
If \(F(k, x, u(k)) = f(x)\) for all \(k\) and \(x\), every trajectory (definition 105) on \([k_0, \infty )\) satisfies \(x(k_0 + n) = f^{n}(x(k_0))\) for every \(n \in \mathbb {N}\).
Induction on \(n\).
The autonomous system (definition 99) \(x^{+} = F(k, x, 0)\), \(y = h(k, x, 0)\) obtained by holding the input at \(0\). Stability of a system with input means stability of this system.
A sequence is a trajectory of the unforced system (definition 112) on \(I\) exactly when it is a trajectory of \(s\) under the zero input on \(I\).