ag-gipp/GoUldI

<math xmlns='http://www.w3.org/1998/Math/MathML' display='inline'><semantics><mrow><munder><munder><mrow><msub><mi>u</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo>,</mo><msub><mi>z</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><msub><mover><mi>u</mi><mo>˙</mo></mover> <mi>x</mi></msub></mrow><mo>⏟</mo></munder><mrow><mtext>By definition of </mtext><msub><mi>v</mi> <mn>1</mn></msub></mrow></munder><mo>=</mo><msup><mover><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mfrac><mrow><mo>∂</mo><msub><mi>V</mi> <mi>x</mi></msub></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle></mrow></mfrac><msub><mi>g</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo stretchy="false">)</mo><mo>−</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>z</mi> <mn>1</mn></msub><mo>−</mo><msub><mi>u</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow></munder><mo stretchy="false">)</mo></mrow><mo>⏞</mo></mover> <mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow></msup><mspace width="thinmathspace"/><mo>+</mo><mspace width="thinmathspace"/><msup><mover><mrow><mfrac><mrow><mo>∂</mo><msub><mi>u</mi> <mi>x</mi></msub></mrow><mrow><mo>∂</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle></mrow></mfrac><mo stretchy="false">(</mo><munder><munder><mrow><msub><mi>f</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo stretchy="false">)</mo><mo>+</mo><msub><mi>g</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo stretchy="false">)</mo><msub><mi>z</mi> <mn>1</mn></msub></mrow><mo>⏟</mo></munder><mrow><mover><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle><mo>˙</mo></mover><mtext> (i.e., </mtext><mfrac><mrow><mi><mi>d</mi></mi><mstyle mathvariant="bold"><mi mathvariant="bold">x</mi></mstyle></mrow><mrow><mi><mi>d</mi></mi><mi>t</mi></mrow></mfrac><mtext>)</mtext></mrow></munder><mo stretchy="false">)</mo></mrow><mo>⏞</mo></mover> <mrow><msub><mover><mi>u</mi><mo>˙</mo></mover> <mi>x</mi></msub><mtext> (i.e., </mtext><mfrac><mrow><mi><mi>d</mi></mi><mi>u</mi><msub><mo/><mi>x</mi></msub></mrow><mrow><mi><mi>d</mi></mi><mi>t</mi></mrow></mfrac><mtext>)</mtext></mrow></msup></mrow><annotation encoding='application/x-tex'>\underbrace{u_1(\mathbf{x},z_1)=v_1+\dot{u}_x}_{\text{By definition of }v_1}=\overbrace{-\frac{\partial V_x}{\partial \mathbf{x}}g_x(\mathbf{x})-k_1(\underbrace{z_1-u_x(\mathbf{x})}_{e_1})}^{v_1} \, + \, \overbrace{\frac{\partial u_x}{\partial \mathbf{x}}(\underbrace{f_x(\mathbf{x})+g_x(\mathbf{x})z_1}_{\dot{\mathbf{x}} \text{ (i.e., } \frac{\operatorname{d}\mathbf{x}}{\operatorname{d}t} \text{)}})}^{\dot{u}_x \text{ (i.e., } \frac{ \operatorname{d}u_x }{\operatorname{d}t} \text{)}}</annotation></semantics></math>