1. Introduction
2. Preliminaries
 Every element E of ${\mathcal{T}}_{h}$ is starshaped with respect to a ball of radius bigger or equal to $\rho {h}_{E}$;
 For every element E of ${\mathcal{T}}_{h}$ and every edge s of E, ${h}_{s}\ge \rho {h}_{E}$.
 the ${L}^{2}projection\phantom{\rule{4pt}{0ex}}{\mathsf{\Pi}}_{n}^{0}:{L}^{2}\left(E\right)\to {\mathbb{P}}_{n}\left(E\right)$, defined by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {(q,v)}_{0,E}={(q,{\mathsf{\Pi}}_{n}^{0}v)}_{0,E}\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4pt}{0ex}}all\phantom{\rule{4pt}{0ex}}v\in {L}^{2}\left(E\right)\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}q\in {\mathbb{P}}_{n}\left(E\right),\end{array}\end{array}$$
 the ${H}^{1}projection\phantom{\rule{4pt}{0ex}}{\mathsf{\Pi}}_{n}^{1}:{H}^{1}\left(E\right)\to {\mathbb{P}}_{n}\left(E\right)$, defined by$$\begin{array}{c}\hfill \begin{array}{c}\hfill {(\nabla q,\nabla v)}_{0,E}={(\nabla q,\nabla {\mathsf{\Pi}}_{n}^{1}v)}_{0,E}\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4pt}{0ex}}all\phantom{\rule{4pt}{0ex}}v\in {H}^{1}\left(E\right)\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}q\in {\mathbb{P}}_{n}\left(E\right)\end{array}\end{array}$$$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int}_{\partial E}(v{\mathsf{\Pi}}_{n}^{1}v)ds=0.\end{array}\end{array}$$
3. SUPG Stabilizing Virtual Element Approximation for Optimal Control Problem
4. A Priori Error Estimates
5. Numerical Results
Algorithm 1: Projected gradient algorithm. 
Require: Regularization parameter $\gamma $ and tolerance error $\eta $. Ensure:

6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Mesh  h  ${\mathit{r}}_{1}$  ${\mathit{r}}_{2}$  ${\mathit{r}}_{3}$  ${\mathit{c}}_{\mathit{y}}$  Rate  ${\mathit{c}}_{\mathit{p}}$  Rate 

Distorted square mesh  $0.131$  $0.193$  $6.225\times {10}^{7}$  $0.800$  $3.097\times {10}^{1}$  $1.237\times {10}^{1}$  
$0.098$  $0.153$  $3.412\times {10}^{7}$  $0.800$  $2.049\times {10}^{1}$  $1.44$  $8.197\times {10}^{2}$  $1.43$  
$0.049$  $0.081$  $8.531\times {10}^{6}$  $0.800$  $7.737\times {10}^{2}$  $1.41$  $3.088\times {10}^{2}$  $1.41$  
$0.033$  $0.054$  $3.892\times {10}^{6}$  $0.800$  $4.114\times {10}^{2}$  $1.56$  $1.641\times {10}^{2}$  $1.56$  
$0.022$  $0.036$  $1.729\times {10}^{6}$  $0.800$  $2.119\times {10}^{2}$  $1.64$  $8.424\times {10}^{3}$  $1.64$  
Hexagonal mesh  $0.125$  $0.208$  $6.406\times {10}^{7}$  $0.800$  $3.852\times {10}^{1}$  $1.522\times {10}^{1}$  
$0.063$  $0.104$  $1.602\times {10}^{7}$  $0.800$  $1.533\times {10}^{1}$  $1.33$  $6.200\times {10}^{2}$  $1.30$  
$0.042$  $0.069$  $7.118\times {10}^{6}$  $0.800$  $8.804\times {10}^{2}$  $1.37$  $3.557\times {10}^{2}$  $1.37$  
$0.036$  $0.059$  $5.230\times {10}^{6}$  $0.800$  $7.107\times {10}^{2}$  $1.40$  $2.864\times {10}^{2}$  $1.40$  
$0.031$  $0.052$  $3.996\times {10}^{6}$  $0.800$  $5.894\times {10}^{2}$  $1.41$  $2.375\times {10}^{2}$  $1.41$  
Lloyd mesh  $0.201$  $0.274$  $2.136\times {10}^{8}$  $0.800$  $7.215\times {10}^{1}$  $2.954\times {10}^{1}$  
$0.097$  $0.158$  $5.442\times {10}^{7}$  $0.800$  $2.539\times {10}^{1}$  $1.44$  $1.005\times {10}^{1}$  $1.49$  
$0.070$  $0.116$  $2.603\times {10}^{7}$  $0.800$  $1.507\times {10}^{1}$  $1.55$  $6.047\times {10}^{2}$  $1.51$  
$0.049$  $0.073$  $1.274\times {10}^{7}$  $0.800$  $9.224\times {10}^{2}$  $1.41$  $3.691\times {10}^{2}$  $1.42$  
$0.035$  $0.052$  $6.429\times {10}^{6}$  $0.800$  $5.382\times {10}^{2}$  $1.66$  $2.145\times {10}^{2}$  $1.67$ 
Mesh  ${\mathit{e}}_{\mathit{y},0}$  Rate  ${\mathit{e}}_{\mathit{y},1}$  Rate  ${\mathit{e}}_{\mathit{p},1}$  Rate  ${\mathit{e}}_{\mathit{p},0}$  Rate  ${\mathit{e}}_{\mathit{u},0}$  Rate 

Distorted square mesh  $3.425\times {10}^{2}$  $5.752\times {10}^{1}$  $2.277\times {10}^{1}$  $1.355\times {10}^{2}$  $8.982\times {10}^{3}$  
$1.952\times {10}^{2}$  $1.95$  $4.289\times {10}^{1}$  $1.02$  $1.670\times {10}^{1}$  $1.08$  $7.238\times {10}^{3}$  $2.18$  $4.787\times {10}^{3}$  $2.19$  
$4.374\times {10}^{3}$  $2.16$  $2.039\times {10}^{1}$  $1.07$  $8.044\times {10}^{2}$  $1.05$  $1.548\times {10}^{3}$  $2.23$  $1.016\times {10}^{3}$  $2.24$  
$1.783\times {10}^{3}$  $2.21$  $1.321\times {10}^{1}$  $1.07$  $5.259\times {10}^{2}$  $1.05$  $6.499\times {10}^{4}$  $2.14$  $4.235\times {10}^{4}$  $2.16$  
$7.613\times {10}^{4}$  $2.10$  $8.287\times {10}^{2}$  $1.15$  $3.304\times {10}^{2}$  $1.15$  $2.679\times {10}^{4}$  $2.18$  $1.715\times {10}^{4}$  $2.23$  
Hexagonal mesh  $3.910\times {10}^{2}$  $7.071\times {10}^{1}$  $2.869\times {10}^{1}$  $1.780\times {10}^{2}$  $1.247\times {10}^{2}$  
$7.108\times {10}^{3}$  $2.46$  $3.388\times {10}^{1}$  $1.06$  $1.368\times {10}^{1}$  $1.07$  $3.470\times {10}^{3}$  $2.36$  $2.213\times {10}^{3}$  $2.49$  
$2.851\times {10}^{3}$  $2.25$  $2.268\times {10}^{1}$  $0.99$  $9.127\times {10}^{2}$  $1.00$  $1.427\times {10}^{3}$  $2.19$  $9.084\times {10}^{4}$  $2.20$  
$2.146\times {10}^{3}$  $1.84$  $1.949\times {10}^{1}$  $0.98$  $7.840\times {10}^{2}$  $0.99$  $1.042\times {10}^{3}$  $2.04$  $6.621\times {10}^{4}$  $2.05$  
$1.678\times {10}^{3}$  $1.86$  $1.710\times {10}^{1}$  $0.99$  $6.870\times {10}^{2}$  $1.00$  $7.992\times {10}^{4}$  $2.00$  $5.072\times {10}^{4}$  $2.01$  
Lloyd mesh  $9.223\times {10}^{2}$  $1.084\times {10}^{0}$  $4.440\times {10}^{1}$  $3.715\times {10}^{2}$  $2.365\times {10}^{2}$  
$1.853\times {10}^{2}$  $2.21$  $4.744\times {10}^{1}$  $1.14$  $1.900\times {10}^{1}$  $1.17$  $7.854\times {10}^{3}$  $2.14$  $4.781\times {10}^{3}$  $2.20$  
$8.421\times {10}^{3}$  $2.35$  $3.303\times {10}^{1}$  $1.08$  $1.325\times {10}^{1}$  $1.07$  $3.661\times {10}^{3}$  $2.27$  $2.227\times {10}^{3}$  $2.28$  
$4.151\times {10}^{3}$  $2.03$  $2.328\times {10}^{1}$  $1.01$  $9.237\times {10}^{2}$  $1.04$  $1.772\times {10}^{3}$  $2.09$  $1.139\times {10}^{2}$  $1.93$  
$2.130\times {10}^{3}$  $2.06$  $1.729\times {10}^{1}$  $0.92$  $6.821\times {10}^{2}$  $0.93$  $9.518\times {10}^{4}$  $1.92$  $6.057\times {10}^{4}$  $1.95$ 
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