--- title: 'Maximal Overshoot: A Cross-disciplinary Overview' url: https://www.emergentmind.com/topics/maximal-overshoot type: topic --- # Maximal Overshoot: A Cross-disciplinary Overview Maximal overshoot is a domain-specific extremal notion for the largest excursion beyond a distinguished level. In the cited arXiv literature, that level may be the post-peak susceptible fraction in an epidemic, the classical Schwarzschild boundary of a convective core, the envelope reconstructed between FFT samples, the physical bounds of a square wave under shock-capturing discretization, the reference trajectory of a feedback loop, or the slow-roll plateau reached after Coleman–De Luccia tunneling. The shared mathematical pattern is an optimization or bounding problem over trajectories, signals, or admissible models, but the state variable and physical interpretation of “overshoot” differ substantially across fields [2303.01650][1404.6153][1602.08975][2102.09876][1103.4198][2006.13914][1109.5182][2203.08686]. ## 1. Cross-disciplinary usage and formal definitions The term does not denote a single universal observable. Instead, each literature fixes a baseline and then studies either the exact maximum attainable excess, the minimum unavoidable excess, or the maximal admissible extension beyond a formal boundary. | Domain | Formal definition | Extremal statement | |---|---|---| | Epidemic dynamics | $\mathrm{Overshoot}=S(t^*)-S_\infty=\frac{1}{R_0}-S_\infty$ | $\mathrm{Overshoot}^*\approx0.2984$ at $R_0^*\approx2.151$ | | Convective-core modeling | $\delta_{ov}=\beta H_P$ or $D_{\rm ov}(r)=D_0\exp\!\bigl[-2|r-r_{\rm cz}|/(f_{\rm ov}H_P)\bigr]$ | Procyon: $\delta_{ov,\max}\simeq1.5\,H_P$; solar-like stars: $f_{\rm ov}^{\rm(Herwig)}\lesssim6\times10^{-3}$ for solar conditions | | Bandlimited signals | $C_1(L)=\sup_{\|f\|_{t_{1,L},\infty}\le1}\|f\|_\infty$ | finite-sum upper bounds for $C_1(L)$ and Nyquist-filter peak-regrowth | | Shock-capturing schemes | $\mathrm{Overshoot}(\phi,\nu)=\max_x u_\epsilon^{\rm overshoot}(x;\phi,\nu)$ | scheme-dependent worst-case $M_S=\max_{\phi,\nu}\mathrm{Overshoot}_S(\phi,\nu)$ | | Continuous-time and governed control | $f_{os}(e)=\sup_{t\ge0}[-e(t)]_+$ or $\sup_{k>t}[y_{tr}(k)-r(t)]\le\delta$ | exact lower bounds from pole/zero data; RG-DC can enforce zero overshoot or a prescribed $\delta$ | | Inflation after tunneling | $\Delta\phi_{\rm CDL}\equiv\phi(t_c)$ | for $n<4$, $\Delta\phi_{\rm CDL}\simeq\frac{3}{4\sqrt2}\sqrt\epsilon+{\cal O}(1)|\phi_0|$; for $n\ge4$, overshooting is entirely absent | A plausible implication is that “maximal overshoot” is best understood as a family of extremal constructions rather than a single transferable metric. The conserved quantities, dual programs, sampling operators, and asymptotic states that enter the calculation are specific to each model class [2303.01650][1602.08975][1103.4198]. ## 2. Epidemic overshoot in the Kermack–McKendrick SIR model In the classical Kermack–McKendrick SIR framework, the population fractions satisfy \[ \frac{dS}{dt}=-\beta SI,\qquad \frac{dI}{dt}=\beta SI-\gamma I,\qquad \frac{dR}{dt}=\gamma I, \] with $S(t)+I(t)+R(t)=1$ and $R_0=\beta/\gamma$. If $t^*=\arg\max_{t\ge0}I(t)$ denotes the time of peak prevalence, then \[ \frac{dI}{dt}\Big|_{t^*}=0 \quad\Longrightarrow\quad S(t^*)=\frac{1}{R_0}. \] The epidemic overshoot is the fraction of the population infected after the peak: \[ \mathrm{Overshoot} =\int_{t^*}^{\infty}\beta S(t)I(t)\,dt =S(t^*)-S_\infty =\frac{1}{R_0}-S_\infty, \] where $S_\infty=\lim_{t\to\infty}S(t)$ [2303.01650]. Using the final-size relation \[ S_\infty=\exp\bigl(R_0(S_\infty-1)\bigr), \qquad R_0=\frac{\ln S_\infty}{S_\infty-1}, \] the overshoot becomes the one-variable function \[ \mathrm{Overshoot}(S_\infty) =\frac{S_\infty-1}{\ln S_\infty}-S_\infty. \] Its critical point satisfies \[ \bigl(\ln S_\infty\bigr)^2=\ln S_\infty-1+\frac{1}{S_\infty}, \] with numerical solution \[ S_\infty^*\approx0.1664,\qquad \mathrm{Overshoot}^*\approx0.2984,\qquad R_0^*\approx2.151. \] Thus, in the unmitigated SIR model, nearly $30\%$ of the population becomes infected after the peak, and the total attack rate at the maximizing point is $1-S_\infty^*\approx0.8336$ [2303.01650]. The same paper extends the analysis to SIR$+$V models. For a fixed pre-outbreak vaccinated fraction $V$, \[ \mathrm{Overshoot}_V^*=(1-V)\,0.2984. \] For risk-driven vaccination $\dot V=\lambda SI$, \[ \mathrm{Overshoot}(\lambda) = \Bigl[\frac{S_\infty-1}{\ln S_\infty}-S_\infty\Bigr] \Bigl(1-\frac{\lambda}{\beta+\lambda}\Bigr) \le 0.2984, \] with equality in the limit $\lambda\to0$. In every scenario considered, the bound $\mathrm{Overshoot}\le0.2984$ remains intact [2303.01650]. The Manaus example gives the notion epidemiological content. An early effective reproduction number $R_0\approx2.3$ implies $\mathrm{Overshoot}_{\mathrm{theory}}\approx29\%$. Seroprevalence data corrected for antibody waning indicate that approximately $36\%$ of the city had already been infected at the peak and about $66\%$ by the end of the wave, yielding an empirical overshoot of roughly $30\%$. The stated interpretation is that simple SIR-model bounds can anticipate the post-peak burden of disease once a sound estimate of $R_0$ becomes available [2303.01650]. ## 3. Convective-core overshoot in stellar structure and asteroseismology In stellar evolution, overshoot refers to mixing beyond the formally convective boundary. Guenther et al. quantify Procyon’s convective-core overshoot by \[ \delta_{ov}=\beta H_P, \] where $H_P\equiv-(dr/d\ln P)$ is the local pressure scale height and $\beta$ ranges from $0\,H_P$ to $2\,H_P$ in steps of $0.1\,H_P$. Two conventional prescriptions were examined. In the raOv models, chemical mixing is uniform over $[r_{cc},r_{cc}+\delta_{ov}]$ while the temperature gradient remains radiative, $\nabla=\nabla_{\rm rad}$. In the adOv models, the same region is forced to be adiabatic, $\nabla=\nabla_{\rm ad}$, mimicking penetrative convection. The Kuhfuss nonlocal theory instead solves a time-dependent turbulent kinetic-energy equation, \[ \partial E_t/\partial t+\nabla\cdot F_t=S_{\rm buoy}+P_{\rm driving}-\epsilon_{\rm diss}, \] and sets the convective boundary by the radius at which the convective velocity vanishes [1404.6153]. The observational constraint is Bayesian evidence from a frequency-fitting framework with residual model \[ f_{\rm obs}-f_{\rm mod} = A\,(f_{\rm obs}/f_{\max})^b+\epsilon, \] where $\epsilon$ is Gaussian and the systematic offset is marginalized using a “$\beta$-prior.” Across all model variants, the evidence rises sharply from $\beta=0$ to $\beta\approx1.0\,H_P$, peaks in the range $\beta\approx0.9$–$1.5\,H_P$, and then gradually declines for $\beta>1.5\,H_P$. In the raOv sequence the maximum evidence occurs near $\beta\approx1.5\,H_P$; in the raOvD sequence it shifts slightly to $\beta\approx1.0\,H_P$. Relative to $\beta=0$, evidence ratios exceed $10^2$, corresponding to posterior odds $>99\%$ in favor of substantial core overshoot. The quoted maximal overshoot is therefore \[ \delta_{ov,\max}\simeq1.5\,H_P \] within the model space explored [1404.6153]. The inferred core enlargement is substantial. Without overshoot, \[ R_{cc}/R\simeq0.059,\qquad M_{cc}/M\simeq0.069. \] With $\delta_{ov}=1.5\,H_P$ these become \[ R_{cc}/R\simeq0.075\quad(+27\%),\qquad M_{cc}/M\simeq0.090\quad(+30\%). \] At $\delta_{ov}=1.0\,H_P$, \[ R_{cc}/R\simeq0.072\quad(+22\%),\qquad M_{cc}/M\simeq0.123\quad(+77\%). \] Guenther et al. explicitly note that these overshoot fractions exceed the “modest” $0.1$–$0.2\,H_P$ often inferred from star-cluster turn-off or eclipsing-binary analyses, and they suggest rotationally induced circulations, tidal mixing from Procyon’s white-dwarf progenitor companion, or shear-driven turbulence as possible effective sources of the larger mixed region. They also state that the Kuhfuss nonlocal model fails to reach as high an evidence as the radiative-gradient prescription, which points to a predominantly over-mixing rather than penetrative mechanism [1404.6153]. A distinct stellar literature constrains overshoot from the absence of a present-day solar convective core. In the exponential-diffusion overshoot model, \[ D_{\rm ov}(r)=C\,D_0\Bigl(\frac{P(r)}{P_{\rm cz}}\Bigr)^\theta, \] or equivalently \[ D_{\rm ov}(r)=D_0\exp\!\biggl[-\frac{2|r-r_{\rm cz}|}{f_{\rm ov}H_P}\biggr], \qquad \theta\approx\frac{2}{f_{\rm ov}}, \qquad C\simeq1. \] Helioseismic inversions require $\delta c/c\lesssim0.1\%$ and $\delta\rho/\rho\lesssim1\%$ for $r<0.3R_\odot$. In the classical step model, a convective core survives to the solar age if $\alpha_{\rm ov}\gtrsim0.25\,H_P$, so one requires \[ \alpha_{\rm ov}\lesssim0.20\mbox{–}0.25 \] for solar conditions. In the exponential-diffusion model with $C=1$, \[ \log_{10}\theta_{\rm cr}\simeq2.53, \qquad f_{\rm ov}\lesssim6\times10^{-3}, \] and this helioseismic limit is stricter than the neutrino-flux constraint [2203.08686]. For $1.01$ one defines $t_{l,L}=\pi l/(LB)$. The sample norm is \[ \|f\|_{t_{1,L},\infty}:=\sup_{l\in\mathbb Z}|f(t_{l,L})|, \] and the maximal overshoot constant is \[ C_1(L) := \sup_{\substack{f\in\mathrm{PW}_B^\infty\\ \|f\|_{t_{1,L},\infty}\le1}} \|f\|_\infty = \sup_{\|f\|_{t_{1,L},\infty}\le1}\sup_{t\in\mathbb R}|f(t)|. \] Equivalently, \[ O(f,L)=\frac{\sup_{t\in\mathbb R}|f(t)|}{\sup_l |f(t_{l,L})|}, \qquad C_1(L)=\sup_{f\neq0}O(f,L). \] This formulation treats overshoot as peak regrowth between samples rather than a state-space excursion [1602.08975]. The same paper embeds the problem in an aperiodic transmit-filter model, \[ x(t)=\sum_{l\in\mathbb Z}X_l\,g\!\Bigl(t-\frac{\pi l}{LB}\Bigr), \] and also in the generalized sampling series \[ f(t)=\sum_{l\in\mathbb Z}f(t_{l,L})\,g(t-t_{l,L}), \qquad g\in\mathcal M_{L_\epsilon}^B,\quad L\ge \frac{L_\epsilon+1}{2}. \] A classical bound due to Litsyn gives \[ C_2(L,L_\epsilon)\le \sqrt{\frac{L_\epsilon+1}{L_\epsilon-1}} \qquad\text{for }L\ge L_\epsilon>1, \] and hence, when $L=L_\epsilon$, \[ C_1(L)\le C_2(L,L)\le \sqrt{\frac{L+1}{L-1}}. \] The paper refines this with a finite-sum estimate for \[ L_\epsilon=\frac{n+1}{n}, \qquad L=\frac{n+m}{n}, \qquad n\in\mathbb N,\; m\in\mathbb N\cup\{\tfrac12\}, \] namely \[ C_2(L,L_\epsilon) \le \max_{t\in\Omega} \frac{1}{2(n+m)} \sum_{l=0}^{2(n+m)-1} \Biggl| \sum_{k=-n}^{n} e^{jk\bigl(\frac{\pi t}{n}-\frac{l\pi}{n+m}\bigr)} \Biggr|, \] with $\Omega=[-n/(2(n+1)),\,n/(2(n+1))]$. Since $C_1(L)\le C_2(L,L_\epsilon)$, this is also a direct upper bound on the maximal overshoot [1602.08975]. The extension to Nyquist filters is significant because classical bounds require $L\ge L_\epsilon$, whereas a Nyquist-criterion filter at half-rate $\omega_N=LB$ necessarily has $L\le L_\epsilon$. The new estimate remains finite even when $L\tfrac12,\\ 0, & \text{otherwise}, \end{cases} \] and the scalar metric is \[ \mathrm{Overshoot}(\phi,\nu)=\max_x u_\epsilon^{\rm overshoot}(x;\phi,\nu). \] This is an exact one-step error metric rather than an asymptotic norm [2102.09876]. The derivation uses the Fourier expansion of the square wave, a semi-discrete symbol $\sigma(\phi)$, and the SSP-RK3 stability function \[ R(z)=1+z+\frac{z^2}{2}+\frac{z^3}{6}. \] For odd Fourier modes, the amplification factor is \[ G(m\phi,\nu)=R(\nu\,\sigma(m\phi)), \] so the pointwise error becomes \[ u_{\epsilon,j} = \sum_{m=0}^\infty \frac{2}{\pi(2m+1)} \Bigl\{ G((2m+1)\phi,\nu)-e^{-i(2m+1)\nu\phi} \Bigr\} \sin[(2m+1)\phi\,j]. \] The overshoot is then extracted by the physical-bound test above and maximized over $j$ [2102.09876]. Two empirical findings are emphasized. First, the overshoot amplitude varies non-monotonously with the CFL number $\nu$; there is not a simple “more CFL $\Rightarrow$ more overshoot” law, and several WENO-type schemes attain minimum overshoot around $\nu\approx0.6$. For 5th-order WENO-Z, the minimum over $\nu\in[0.1,0.9]$ occurs at $\nu\approx0.65$ and is approximately $70\%$ of its value at $\nu=0.4$. Second, overshoot depends strongly on the reduced wavenumber $\phi$: there is a threshold $\phi_0\approx1.05$ below which all schemes produce essentially zero overshoot, while for $\phi>\phi_0$ the overshoot grows, often in a staircase fashion, and some schemes exhibit local maxima and minima in $\phi$ [2102.09876]. The worst-case survey over $\phi\in[0,\pi]$ and $\nu\in(0,0.9]$ gives the following approximate maxima: | Scheme | $M_S$ | |---|---:| | TVD (minmod) | $0$ | | TVD (superbee) | $\approx2\times10^{-3}$ | | ENO3 | $0$ | | ENO5 | $\approx1.5\times10^{-2}$ | | WENO-JS5 | $\approx1.2\times10^{-1}$ | | WENO-JS7 | $\approx1.8\times10^{-1}$ | | WENO-Z5 | $\approx3.0\times10^{-1}$ | | WENO-Z7 | $\approx3.5\times10^{-1}$ | | MPWENO5 | $\approx4.0\times10^{-2}$ | | MPWENO7 | $\approx5.0\times10^{-2}$ | The paper attributes these rankings to strict non-oscillatory behavior for TVD(minmod) and ENO3, Gibbs-type over/undershoots when higher-order ENO/WENO stencils span discontinuities, and reduced dissipation but larger linear-stencil oscillations in WENO-Z. The monotonicity-preserving post-limiter in MPWENO greatly curtails these oscillations [2102.09876]. ## 6. Overshoot in continuous-time control theory and reference governance In continuous-time feedback systems, Wenczel and Hill formulate minimum achievable overshoot as a Banach-space optimization. For a plant \[ P(s)=\hat n(s)/\hat d(s)\in\mathcal D \] with unstable poles $p_1,\dots,p_m\in\mathbb C_+$ and nonminimum-phase zeros $z_1,\dots,z_n\in\mathbb C_+$, the attainable rational error signals form \[ \mathcal F = \left\{ e\in C_0([0,\infty)) \;\middle|\; \hat e \text{ rational},\; \hat e(z_i)=\hat w(z_i),\; \hat e(p_j)=0 \right\}. \] The overshoot functional is \[ f_{os}(e)=\sup_{t\ge0}[-e(t)]_+, \qquad [x]_+=\max\{0,x\}, \] and the primal problem is \[ \min_{e\in X_\theta,\;Ae=b} f_{os}(e). \] Fenchel duality yields the OS-dual \[ \sup_{\mu\in U\oplus V}\langle \mu,w\rangle \quad\text{s.t.}\quad \mu\le0,\;\|\mu\|\le1, \] or, in density form, \[ \max_{e^*\in U\oplus V} \int_0^\infty e^*(t)\,w(t)\,dt \quad\text{s.t.}\quad e^*(t)\le0,\; \int_0^\infty |e^*(t)|\,dt\le1. \] The paper states strong duality: the minimum overshoot equals the maximum dual value [1103.4198]. The key pole/zero sensitivity theorem isolates the dominant right-half-plane locations. If there is a nonminimum-phase zero $z\in\mathbb R_{>0}$ and an unstable pole $p\in\mathbb R_{>0}$ with $z>p$, then for a unit step \[ \boxed{\OS_{\min}=\frac{p}{z-p}}. \] If there are no unstable poles but a single RHP zero $z>0$, then \[ \OS_{\min}=0. \] For the illustrative plant \[ P(s)=\frac{s-3}{s-1}, \] the bound is $\OS_{\min}=0.5$, and the paper reports that time-domain simulation with a high-bandwidth rational stabilizing controller produces overshoot no smaller than about $0.52$, confirming that the bound $0.50$ is tight [1103.4198]. A different control literature addresses overshoot mitigation rather than fundamental impossibility. The Reference Governor with Dynamic Constraint (RG-DC) starts from the asymptotically stable discrete-time model \[ x(t+1)=A\,x(t)+B\,v(t), \] \[ y_{tr}(t)=C_{tr}x(t)+D_{tr}v(t), \qquad y_{st}(t)=C_{st}x(t)+D_{st}v(t), \] with static output constraints $y_{st}(t)\in Y_{st}=\{y:S_{st}y\le s_{st}\}$. The dynamic tracking constraint is \[ y_{tr}(k)\in Y_{tr}(r,y_{tr}^0),\quad \forall k\ge0, \] where $Y_{tr}(r,y_{tr}(t))=\{y:y\le r\}$ if $y_{tr}(t)\le r$, and $\{y:y\ge r\}$ otherwise. The corresponding dynamic maximal admissible set is \[ M(r,y_{tr}^0)\equiv O_{\infty,tr}(r,y_{tr}^0), \] defined by constant future command $v(k)=v$ and simultaneous satisfaction of $y_{st}(k)\in Y_{st}$ and $y_{tr}(k)\in Y_{tr}(r,y_{tr}^0)$ for all $k\ge0$ [2006.13914]. The online law computes \[ r_g(t)\equiv v(t)=v(t-1)+\kappa^*[r(t)-v(t-1)], \] with $\kappa^*$ chosen so that the current pair $(x(t),v(t))$ remains inside the time-varying polyhedral MAS. In the basic RG-DC, \[ \sup_{k>t}\bigl(y_{tr}(k)-r(t)\bigr)\le0, \] so the maximal overshoot above the current reference is zero. If one permits up to $\delta>0$ of overshoot by replacing $Y_{tr}(r)$ with $\{y:y\le r+\delta\}$, the same construction enforces \[ \sup_{k>t}[y_{tr}(k)-r(t)]\le\delta. \] The paper further states invariance, recursive feasibility, BIBO stability, and convergence for constant references, and observes that RG-DC completely suppresses the resonant peak of an under-damped loop, acting like a novel anti-resonant low-pass filter [2006.13914]. ## 7. Overshoot in inflation after Coleman–De Luccia tunneling Dutta, Vaudrevange, and Westphal study overshoot of the inflaton after CDL tunneling in an open FRW universe. For monomial exit potentials \[ V(\phi)=\frac{\lambda_n}{n}\phi^n,\qquad n<4, \] with $\phi(0)=\phi_0<0$ and $\dot\phi(0)=0$, curvature dominates at early times, $a(t)\approx t$ and $H\approx1/t$, so the field obeys \[ \ddot\phi+\frac{3}{t}\dot\phi+(-1)^n\lambda_n\phi^{n-1}=0. \] After reaching $\phi=0$ it enters the plateau \[ V(\phi)\simeq V_-(1-\sqrt{2\epsilon}\,\phi), \qquad t_c\simeq\sqrt{3/V_-}, \] and the overshoot distance is \[ \Delta\phi_{\rm CDL}\equiv \phi(t_c) \simeq \frac{3}{4\sqrt2}\sqrt\epsilon+{\cal O}(1)|\phi_0|. \] The coefficient of $\sqrt\epsilon$ is universal for $n=1,2,3$, while the ${\cal O}(1)$ term decreases with $n$ [1109.5182]. The paper gives explicit formulas and bounds for the low-power cases. For $n=1$, \[ \phi(t_c)=\frac{3}{4\sqrt2}\sqrt\epsilon-\phi_0. \] For $n=2$, \[ \phi(t_c)\simeq0.53\sqrt\epsilon-0.403\,\phi_0. \] For $n=3$, \[ \phi(t_c)\in \Bigl[ \frac{3}{4\sqrt2}\sqrt\epsilon-\frac13\phi_0,\; \frac{3}{4\sqrt2}\sqrt\epsilon-0.1\,\phi_0 \Bigr]. \] The broad statement is that for $n<4$ the overshoot is bounded from above by the width of the steep barrier traversed after emerging from tunneling and before reaching a slow-roll region of the potential [1109.5182]. For $n\ge4$ the behavior changes qualitatively. At $n=4$ the exact solution is \[ \phi(t)=\frac{8\phi_0}{8+\lambda_4\phi_0^2 t^2}, \] for which $\dot\phi\to0$ as $t\to\infty$. More generally, for any monomial with $n\ge4$, the combination of a steeper outer potential and Hubble friction $\sim1/t$ brings $\dot\phi\to0$ before or exactly at $\phi=0$, so overshooting is entirely absent [1109.5182]. The result extends to binomials and then to full power series, \[ V(\phi)=(-1)^m\frac{\lambda_m}{m}\phi^m+(-1)^n\frac{\lambda_n}{n}\phi^n,\qquad n>m, \] with matching point \[ \phi_*\simeq (\lambda_m/\lambda_n)^{1/(n-m)}. \] The overshoot is again controlled by the lowest non-zero monomial above the plateau, \[ \Delta\phi_{\rm CDL}\simeq \frac{3}{4\sqrt2}\sqrt\epsilon+{\cal O}(1)|\phi_*|. \] If one allows arbitrary finite initial speed after tunneling, the early-time equation \[ \ddot\phi+\frac{3}{t}\dot\phi\approx0 \] implies rapid redshifting of $\dot\phi(t)$, so the same bounds remain valid. The stated cosmological implication is that, in a landscape populated by CDL tunneling, small-field models suffer no parametrically large overshoot, and for exit potentials of order $n\ge4$ they have zero classical overshoot [1109.5182]. Source: https://www.emergentmind.com/topics/maximal-overshoot